Powers and Roots: True / false

In order to determine the correctness (or incorrectness) of the given equation, let's simplify the expressions on both sides separately:

A. Let's start with the expression on the left side:
$\sqrt{36}-(4^2-9)+\sqrt{4}$Let's simplify this expression while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and parentheses come before all, therefore we'll start by simplifying the expression in parentheses by calculating the numerical value of the term with the exponent inside them, then we'll perform the subtraction operation in the parentheses:

$\sqrt{36}-(4^2-9)+\sqrt{4} =\\ \sqrt{36}-(16-9)+\sqrt{4} =\\ \sqrt{36}-7+\sqrt{4}$Next, we'll calculate the numerical value of the roots in the expression (which are exponents in every way) and finally we'll perform the result of the expression combining addition and subtraction:

$\sqrt{36}-7+\sqrt{4}=\\ 6-7+2=\\ 1$We have completed simplifying the expression on the left side of the given equation, let's summarize the simplification process:

$\sqrt{36}-(4^2-9)+\sqrt{4} =\\ 6-7+2=\\ 1$

B. Let's continue with simplifying the expression on the right side of the given equation:

$\sqrt{\frac{25}{10000}}+\frac{95}{100}$For this, let's recall two laws of exponents:

B.1. The definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$B.2. The law of exponents for an exponent applied to parentheses containing a product of terms:

$\big(\frac{a}{c}\big)^n=\frac{a^n}{c^n}$Unlike in previous questions, we will not convert the square root to a power of one-half, but rather understand and internalize that according to the law of exponents mentioned in B.1. - a root is an exponent in every way and therefore all laws of exponents apply to it,

Let's return to the expression in question and apply this understanding to the first term on the left, which is the term with the root. Note that in this term, the power of one-half (meaning - the exponent equivalent to the square root) applies to the entire fraction under the root, therefore despite the absence of parentheses in the expression, we'll treat the fraction under the root as a fraction within parentheses with the power of one-half (of the root) applied to it, and therefore we'll apply the law of exponents mentioned in B.2. to this term, meaning - we'll apply the root to both the numerator and denominator of the fraction:

$\sqrt{\frac{25}{10000}}+\frac{95}{100} =\\ \frac{\sqrt{25}}{\sqrt{10000}}+\frac{95}{100}$Let's continue and calculate the numerical value of the roots in the numerator and denominator of the fraction, then perform the addition operation between the fractions and simplify the resulting expression:

$\frac{\sqrt{25}}{\sqrt{10000}}+\frac{95}{100} =\\ \frac{5}{100}+\frac{95}{100} =\\ \frac{5+95}{100}=\\ \frac{100}{100} =\\ 1$We performed the addition of fractions directly by putting them on one fraction line and adding the numerators (since the denominators in both fractions are identical, it is the common denominator, so there was no need to expand them), then we used the fact that dividing any number by itself always gives the result 1.

We have completed simplifying the expression on the right side of the given equation, let's summarize the simplification process:

$\sqrt{\frac{25}{10000}}+\frac{95}{100} =\\ \frac{\sqrt{25}}{\sqrt{10000}}+\frac{95}{100} =\\ \frac{5}{100}+\frac{95}{100} =\\ 1$

Let's now return to the equation given in the problem and substitute the expressions on the left and right sides with the results of the simplifications detailed in A and B above, in order to determine the correctness (or incorrectness) of the given equation:

$\sqrt{36}-(4^2-9)+\sqrt{4}=\sqrt{\frac{25}{10000}}+\frac{95}{100} \\ \downarrow\\ 1=1$ We can now definitively determine that the given equation is indeed correct, meaning we have a true statement,

Therefore the correct answer is answer A.

To determine if the given equation is correct, we will simplify each expression in its sides separately,

This will be done while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

A. Let's start with the expression on the left side of the given equation:

$3^4:(\sqrt{25}+2^2)-2^3:\sqrt{4}$We'll begin by simplifying the expressions inside the parentheses, this is done by calculating the numerical value of the terms with exponents (while remembering the definition of a root as an exponent stating that a root is actually an exponent), simultaneously we'll calculate the numerical value of the terms with exponents in the second term from the left and in the first term from the left, which is the divisor in this term (in fact, if we were to convert this expression to a simple fraction, this term would be in the fraction's numerator):

$3^4:(\sqrt{25}+2^2)-2^3:\sqrt{4} =\\ 81:(5+4)-8:2$We'll continue and finish simplifying the expression inside the parentheses, meaning we'll perform the addition operation within them, then we'll remember again the order of operations, meaning that division comes before subtraction and therefore we'll perform the division operations in both terms of the expression, in the final stage we'll perform the subtraction operation:

$81:(5+4)-8:2 =\\ 81:9-8:2 =\\ 9-4=\\ 5$We have finished simplifying the expression on the left side of the given equation, let's summarize the simplification steps:

$3^4:(\sqrt{25}+2^2)-2^3:\sqrt{4} =\\ 81:(5+4)-8:2 =\\ 9-4=\\ 5$

B. Let's continue with simplifying the expression on the right side of the given equation:

$3^4:\sqrt{25}+(2^2-2^3):\sqrt{4}$We'll start again by simplifying the expressions inside the parentheses, this is done by calculating the numerical value of the terms with exponents in the parentheses, simultaneously we'll calculate the numerical value of the other terms with exponents in the given expression (while remembering the definition of a root as an exponent stating that a root is actually an exponent):

$3^4:\sqrt{25}+(2^2-2^3):\sqrt{4} =\\ 81:5+(4-8):2$We'll continue and finish simplifying the expression inside the parentheses, meaning we'll perform the subtraction operation within them, then we'll remember again the order of operations, meaning that division comes before addition and therefore we'll perform the division operations in both terms of the expression, in the final stage we'll perform the addition operation:

$81:5+(4-8):2 =\\ 81:5+(-4):2 =\\ 81:5-4:2=\\ 20\frac{1}{5}-2\\ 18\frac{1}{5}$Note that the result of the subtraction operation in the parentheses yielded a negative result and therefore in the next step we kept this result in parentheses and then applied the multiplication law stating that multiplying a positive number by a negative number gives a negative result (so ultimately we get a subtraction operation), then, since the division operation performed in the first term from the left yielded a non-whole result (actually greater than a whole number) we wrote this result as a mixed fraction,

We have finished simplifying the expression on the right side of the given equation, let's summarize the simplification steps:

$3^4:\sqrt{25}+(2^2-2^3):\sqrt{4} =\\ 81:5+(4-8):2 =\\ 81:5-4:2=\\ 18\frac{1}{5}$Let's now return to the given equation and substitute in its sides the results of simplifying the expressions detailed in A and B:

$3^4:(\sqrt{25}+2^2)-2^3:\sqrt{4}=3^4:\sqrt{25}+(2^2-2^3):\sqrt{4} \\ \downarrow\\ 5=18\frac{1}{5}$Obviously this equation does not hold, meaning - we got a false statement,

Therefore the correct answer is answer B.

In order to determine if the given equation is correct, we will simplify each of the expressions on its sides separately,

This will be done while adhering to the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

A. Let's start with the expression on the left side of the given equation:

$4^3+(\sqrt{49}+\sqrt{64})+2^2$We'll begin by simplifying the expressions inside the parentheses, this is done by calculating the numerical value of the terms with exponents (while remembering the definition of a root as an exponent stating that a root is actually an exponent), simultaneously we'll calculate the numerical value of the terms with exponents, both the third term from the left and the first term from the left:

$4^3+(\sqrt{49}+\sqrt{64})+2^2 =\\ 64+(7+8)+4=\\$Note that the parentheses in this problem have no significance, as all operations between the different numerical terms are addition operations, however, parentheses determine operation precedence and therefore we'll first complete calculating the expression inside the parentheses and only then perform the addition operations (of course, in this case, this order of operations will yield the same result as if we removed the parentheses and calculated the sum of all terms):

$64+(7+8)+4=\\ 64+15+4=\\ 83$We have finished simplifying the expression on the left side of the given equation, let's summarize the simplification steps:

$4^3+(\sqrt{49}+\sqrt{64})+2^2 =\\ 64+(7+8)+4=\\ 83$B. Let's continue with simplifying the expression on the right side of the given equation:

$(4^3+\sqrt{49})+\sqrt{64}+2^2$We'll begin again by simplifying the expressions inside the parentheses, this is done by calculating the numerical value of the terms with exponents (while remembering the definition of a root as an exponent stating that a root is actually an exponent), simultaneously we'll calculate the numerical value of the terms with exponents, both the third term from the left and the second term from the left:

$(4^3+\sqrt{49})+\sqrt{64}+2^2 =\\ (64+7)+8+4$Similar to the previous part, note that the parentheses in this problem have no significance, as all operations between the different numerical terms are addition operations, however, again we'll point out that parentheses determine operation precedence and therefore we'll first complete calculating the expression inside the parentheses and only then perform the addition operations:

$(64+7)+8+4 =\\ 71+8+4=\\ 83$We have finished simplifying the expression on the right side of the given equation, let's summarize the simplification steps:

$(4^3+\sqrt{49})+\sqrt{64}+2^2 =\\ (64+7)+8+4 =\\ 83$$$Let's now return to the given equation and substitute in its sides the results of simplifying the expressions detailed in A and B:

$4^3+(\sqrt{49}+\sqrt{64})+2^2=(4^3+\sqrt{49})+\sqrt{64}+2^2 \\ \downarrow\\ 83=83$We found that this equation indeed holds true, meaning - we got a true statement,

Therefore the correct answer is answer A.

In order to determine if the given equation is correct, we will simplify each of the expressions on its sides separately,

This will be done while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

A. Let's start with the expression on the left side of the given equation:

$4^3-(\sqrt{49}+\sqrt{64})\cdot2^2$We'll begin by simplifying the expressions inside the parentheses, this is done by calculating the numerical values of the terms with exponents (while remembering the definition of a root as an exponent, which states that a root is actually an exponent), simultaneously we'll calculate the numerical value of the term with the exponent, which is the multiplier to the right of the parentheses in the second expression from the left and the numerical value of the term with the exponent - the first from the left:

$4^3-(\sqrt{49}+\sqrt{64})\cdot2^2 =\\ 64-(7+8)\cdot4=\\$We'll continue to perform the addition operation inside the parentheses, in the next step we'll calculate the multiplication by the second term from the left and finally we'll perform the subtraction operation:

$64-(7+8)\cdot4=\\ 64-15\cdot4=\\ 64-60=\\ 4$We have finished simplifying the expression on the left side of the given equation, let's summarize the simplification steps:

$4^3-(\sqrt{49}+\sqrt{64})\cdot2^2 =\\ 64-(7+8)\cdot4=\\ 64-60=\\ 4$B. Let's continue with simplifying the expression on the right side of the given equation:

$(4^3-\sqrt{49})+\sqrt{64}\cdot2^2$Similar to the previous part, we'll start by simplifying the expression in parentheses, this is done by calculating the numerical values of the terms with exponents (and of course this includes the square root), then we'll perform the subtraction operation in the parentheses, simultaneously we'll calculate the numerical values of the root in the second term from the left and of the term with the exponent multiplying it:

$(4^3-\sqrt{49})+\sqrt{64}\cdot2^2 =\\ (64-7)+8\cdot4 =\\ 57 +8\cdot4$We'll continue and perform the multiplication in the second term from the left in the next step we'll perform the addition operation:

$57 +8\cdot4 =\\ 57+32=\\ 89$We have finished simplifying the expression on the right side of the given equation, let's summarize the simplification steps:

$(4^3-\sqrt{49})+\sqrt{64}\cdot2^2 =\\ (64-7)+8\cdot4 =\\ 57+32=\\ 89$Let's now return to the given equation and substitute in its sides the results of simplifying the expressions detailed in A and B:

$4^3-(\sqrt{49}+\sqrt{64})\cdot2^2=(4^3-\sqrt{49})+\sqrt{64}\cdot2^2 \\ \downarrow\\ 4=89$Obviously this equation does not hold true, meaning - we got a false statement,

Therefore the correct answer is answer B.

In order to determine if the given equation is correct, we will simplify each expression on its sides separately,

This will be done while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

A. Let's start with the expression on the left side of the given equation:

$(5^2+3):2^2$First, we'll simplify the expression in parentheses by calculating the numerical value of the exponent term and then perform the addition:

$(5^2+3):2^2 =\\ (25+3):2^2 \\ 28:2^2$Next, we'll calculate the numerical value of the divisor which is a term with an exponent (in fact, if we were to write the division as a fraction, this term would be in the denominator), then we'll perform the division:

$28:2^2 =\\ 28:4=\\ 7$We have completed simplifying the expression on the left side of the given equation, let's summarize the simplification steps,

We got that:

$(5^2+3):2^2 =\\ 28:2^2 =\\ 28:4=\\ 7$

B. Let's continue and simplify the expression on the right side of the given equation:

$5^2+3:2^2$Similar to what we did in the previous part, we'll simplify the expression while following the order of operations,

Note that exponents come before multiplication and division which come before addition and subtraction, so we'll start by calculating the numerical value of the exponent term, which is the second term from the left, and calculate the numerical value of the second exponent term in this expression, which is the first term from the left, then we'll perform the division operation, and finally perform the addition:

$5^2+3:2^2 =\\ 25+3:4=\\ 25+\frac{3}{4} =\\ 25\frac{3}{4}$In the final steps, since the result of the division operation isn't a whole number, we expressed its result as a simple fraction, and the addition result as a mixed number,

We have completed simplifying the expression on the right side of the given equation, this simplification was brief, so there's no need to summarize,

Now let's return to the original equation and substitute the results of simplifying the expressions detailed in A and B:

$(5^2+3):2^2=5^2+3:2^2 \\ \downarrow\\ 7=25\frac{3}{4}$Obviously:

$7\neq25\frac{3}{4}$Therefore the given equation is incorrect, meaning we have a false statement,

Thus the correct answer is answer B.

In order to determine if the given equation is correct, we will simplify each of the expressions in its sides separately,

This will be done while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

A. Let's start with the expression on the left side of the given equation:

$2^3\cdot(6^2-7\cdot3):(\sqrt{36}-\sqrt{1})+\sqrt{4}$We'll start by simplifying the expressions inside the parentheses, this will be done by calculating the numerical values of the terms with exponents (while remembering the definition of a root as an exponent), simultaneously we'll calculate the numerical value of the root in the second term from the left and the numerical value of the term with the exponent, which is the first multiplier from the left in the leftmost expression:

$2^3\cdot(6^2-7\cdot3):(\sqrt{36}-\sqrt{1})+\sqrt{4} =\\ 8\cdot(36-21):(6-1)+2$Let's continue and perform the subtraction operations in the parentheses:

$8\cdot(36-21):(6-1)+2 =\\ 8\cdot15:5+2$Note that there is no defined order of operations between multiplication and division, and there are no parentheses in this expression defining precedence for either operation, therefore we'll calculate the result of the leftmost term (with all its operations) as we compute step by step from left to right, then we'll perform the addition operation:

$8\cdot15:5+2 =\\ 120:5+2 =\\ 24+2=\\26$We have completed simplifying the expression on the left side of the given equation, let's summarize the simplification steps:
$2^3\cdot(6^2-7\cdot3):(\sqrt{36}-\sqrt{1})+\sqrt{4} =\\ 8\cdot(36-21):(6-1)+2 =\\ 8\cdot15:5+2 =\\ 24+2=\\26$

B. Let's continue with simplifying the expression on the right side of the given equation:

$(4^3-5^2):(2+1)\cdot2$

Similar to the previous part, we'll start by simplifying the expressions in parentheses, this will be done by calculating the numerical values of the terms with exponents, then we'll perform the addition and subtraction operations in the parentheses:

$$$(4^3-5^2):(2+1)\cdot2 =\\ (64-25):(2+1)\cdot2=\\ 39:3\cdot2$Note (again) that there is no defined order of operations between multiplication and division, and there are no parentheses in this expression defining precedence for either operation, therefore we'll calculate the result of the expression we got (with all its operations) as we compute step by step from left to right:

$39:3\cdot2 =\\ 13\cdot2=\\ 26$

We have completed simplifying the expression on the right side of the given equation, let's summarize the simplification steps:

$(4^3-5^2):(2+1)\cdot2 =\\ 39:3\cdot2 =\\ 26$ Let's now return to the given equation and substitute in its sides the results of simplifying the expressions detailed in A and B:

$2^3\cdot(6^2-7\cdot3):(\sqrt{36}-\sqrt{1})+\sqrt{4}=(4^3-5^2):(2+1)\cdot2 \\ \downarrow\\ 26=26$We found that the equation is indeed true, meaning - we got a true statement,

Therefore the correct answer is answer A.

To determine if the given equation is correct, we will simplify each expression in its sides separately,

This will be done while adhering to the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

A. Let's start with the expression on the left side of the given equation:

$3^4-(\sqrt{25}-2^2)-(2^3-\sqrt{4})$We'll begin by simplifying the expressions inside the parentheses, this is done by calculating the numerical values of the terms with exponents (while remembering the definition of a root as an exponent stating that a root is actually an exponent), simultaneously we'll calculate the numerical value of the term with the exponent - the leftmost term:

$3^4-(\sqrt{25}-2^2)-(2^3-\sqrt{4}) =\\ 81-(5-4)-(8-2)$We'll continue and finish simplifying the expressions inside the parentheses, meaning we'll perform the subtraction operations in them, then we'll perform the remaining subtraction operation:

$81-(5-4)-(8-2) =\\ 81-1-6=\\ 74$We have finished simplifying the expression on the left side of the given equation, let's summarize the simplification steps:

$3^4-(\sqrt{25}-2^2)-(2^3-\sqrt{4}) =\\ 81-1-6=\\ 74$B. Let's continue with simplifying the expression on the right side of the given equation:

$3^4-\sqrt{25}-(2^2-2^3)-\sqrt{4}$We'll start by simplifying the expression inside the parentheses, this is done by calculating the numerical values of the terms with exponents (while remembering the definition of a root as an exponent stating that a root is actually an exponent), simultaneously we'll calculate the numerical values of the terms with exponents that are not in parentheses:

$3^4-\sqrt{25}-(2^2-2^3)-\sqrt{4} =\\ 81-5-(4-8)-2$We'll continue and finish simplifying the expression inside the parentheses, meaning we'll perform the subtraction operation in it, then we'll perform the remaining subtraction operations:$81-5-(4-8)-2 =\\ 81-5-(-4)-2 =\\ 81-5+4-2 =\\ 78$Note that the result of the subtraction operation in the parentheses yielded a negative result and therefore in the next step we kept this result in parentheses and then applied the multiplication law stating that multiplying a negative number by a negative number gives a positive result (so ultimately we get an addition operation), then, we performed the addition and subtraction operations in the resulting expression,

We have finished simplifying the expression on the right side of the given equation, let's summarize the simplification steps:

$3^4-\sqrt{25}-(2^2-2^3)-\sqrt{4} =\\ 81-5-(-4)-2 =\\ 78$

Let's now return to the given equation and substitute in its sides the results of simplifying the expressions detailed in A and B:

$3^4-(\sqrt{25}-2^2)-(2^3-\sqrt{4})=3^4-\sqrt{25}-(2^2-2^3)-\sqrt{4} \\ \downarrow\\ 74=78$Obviously this equation does not hold, meaning - we got a false statement,

Therefore the correct answer is answer B.

To determine if the given equation is correct, we need to simplify each expression in its sides separately,

This is done while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

A. Let's start with the expression on the left side of the given equation:

$3^4-(\sqrt{25}-2^2)-(2^3-\sqrt{4})$Let's start by simplifying the expressions inside the parentheses, this is done by calculating the numerical values of the terms with exponents (while remembering the definition of a root as an exponent stating that a root is actually an exponent), at the same time we'll calculate the numerical value of the term with the exponent - the leftmost term:

$3^4-(\sqrt{25}-2^2)-(2^3-\sqrt{4}) =\\ 81-(5-4)-(8-2)$Let's continue and finish simplifying the expressions inside the parentheses, meaning we'll perform the subtraction operations in them, then we'll perform the remaining subtraction operation:

$81-(5-4)-(8-2) =\\ 81-1-6=\\ 74$We have finished simplifying the expression on the left side of the given equation, let's summarize the simplification steps:

$3^4-(\sqrt{25}-2^2)-(2^3-\sqrt{4}) =\\ 81-1-6=\\ 74$B. Let's continue with simplifying the expression on the right side of the given equation:

$3^4-\sqrt{25}+(2^2-2^3)+\sqrt{4}$Let's start by simplifying the expression inside the parentheses, this is done by calculating the numerical values of the terms with exponents (while remembering the definition of a root as an exponent stating that a root is actually an exponent), at the same time we'll calculate the numerical values of the terms with exponents that are not in parentheses:

$3^4-\sqrt{25}+(2^2-2^3)+\sqrt{4} =\\ 81-5+(4-8)+2$Let's continue and finish simplifying the expression inside the parentheses, meaning we'll perform the subtraction operation in it, then we'll perform the remaining subtraction operations:$81-5+(4-8)+2 =\\ 81-5+(-4)+2 =\\ 81-5-4+2 =\\ 74$Note that the result of the subtraction operation in the parentheses yielded a negative result and therefore in the next step we kept this result in parentheses and then applied the multiplication law stating that multiplying a positive number by a negative number gives a negative result (so ultimately we get a subtraction operation), then, we performed the subtraction operations in the resulting expression,

We have finished simplifying the expression on the right side of the given equation, let's summarize the simplification steps:

$3^4-\sqrt{25}+(2^2-2^3)+\sqrt{4} =\\ 81-5+(-4)+2 =\\ 74$Let's now return to the given equation and substitute in its sides the results of simplifying the expressions detailed in A and B:

$3^4-(\sqrt{25}-2^2)-(2^3-\sqrt{4})=3^4-\sqrt{25}+(2^2-2^3)+\sqrt{4} \\ \downarrow\\ 74=74$Indeed the equation holds true, meaning - we got a true statement,

Therefore the correct answer is answer A.

To determine if the given equation is correct, we will simplify each of the expressions in its sides separately,

This will be done while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

A. Let's start with the expression on the left side of the given equation:

$5^3-(4^2+3^2)-(\sqrt{100}+8^2)$

We'll start by simplifying the expressions inside the parentheses, we'll do this by calculating the numerical value of the terms with exponents (while remembering the definition of a root as an exponent, meaning that a root is actually an exponent), simultaneously we'll calculate the numerical value of the other terms with exponents in the expression:

$5^3-(4^2+3^2)-(\sqrt{100}+8^2) =\\ 125-(16+9)-(10+64)$

We'll continue and finish simplifying the expressions inside the parentheses, meaning we'll perform the addition operations in them, then we'll perform the remaining subtraction operations:

$125-(16+9)-(10+64) =\\ 125-25-74 =\\ 26$We have finished simplifying the expression on the left side of the given equation, let's summarize the simplification steps:

$5^3-(4^2+3^2)-(\sqrt{100}+8^2) =\\ 125-25-74 =\\ 26$

B. Let's continue with simplifying the expression on the right side of the given equation:

$5^3-4^2-3^2-\sqrt{100}-8^2$

Let's remember again the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these, and note that while this expression has no parentheses, it does have terms with exponents, so we'll start by calculating the numerical value of the terms with exponents, then we'll perform the subtraction operations:

$5^3-4^2-3^2-\sqrt{100}-8^2= \\ 125-16-9-10-64 =\\ 26$We have finished simplifying the expression on the right side of the given equation, this simplification was brief, so there's no need to summarize,

Let's now return to the given equation and substitute in its sides the results of simplifying the expressions that were detailed in A and B:

$5^3-(4^2+3^2)-(\sqrt{100}+8^2)=5^3-4^2-3^2-\sqrt{100}-8^2 \\ \downarrow\\ 26=26$

Of course this equation is indeed true, meaning - we got a true statement,

Therefore the correct answer is answer A.

To determine if the given equality is correct we will simplify each of the expressions that appear in it separately,

This is done while keeping in mind the order of operations which states that multiplication precedes division and subtraction precedes addition and that parentheses precede all,

A. Let's start then with the expression on the left side of the given equality:

$5^3:(4^2+3^2)-(\sqrt{100}-8^2)$We start by simplifying the expressions inside the parentheses, this is done by calculating their numerical value (while remembering the definition of the square root as the non-negative number whose square gives the number under the root), in parallel we calculate the numerical value of the other terms in the expressions:

$5^3:(4^2+3^2)-(\sqrt{100}-8^2) =\\ 125:(16+9)-(10-64)$We continue and finish simplifying the expressions inside the parentheses, meaning we perform the subtraction operation in them, then we perform the division operation which is in the first term from the left and then the remaining subtraction operation:

$125:(16+9)-(10-64) =\\ 125:25-(-54) =\\ 5+54 = 59$We note that the result of the subtraction operation in the parentheses is a negative result and therefore in the next step we will leave this result in the parentheses and then apply the multiplication law which states that multiplying a negative number by a negative number will give a positive result (so that in the end an addition operation is obtained), then, we perform the addition operation in the expression that was obtained,

We finished simplifying the expression on the left side of the given equality, let's summarize the simplification steps:

$5^3:(4^2+3^2)-(\sqrt{100}-8^2) =\\ 125:(16+9)-(10-64) =\\ 5+54 =\\ 59$

B. We continue from simplifying the expression on the right side of the given equality:

$5^3:4^2+3^2-\sqrt{100}+8^2$We recall again the order of operations which states that multiplication precedes division and subtraction precedes addition and that parentheses precede all, and note that although in this expression there are no parentheses, there are terms in fractions and a division operation, so we start by calculating their numerical value, then we perform the division operation:

$5^3:4^2+3^2-\sqrt{100}+8^2 =\\ 125:16+9-10+64 =\\ 7\frac{13}{16}+9-10+64=\\ 70\frac{13}{16}$We note that since the division operation that was performed in the first term from the left yielded an incomplete result (greater than the divisor), we marked this result as a mixed number, then we performed the remaining addition and subtraction operations,

We finished simplifying the expression on the right side of the given equality, the simplification of this expression is short, so there is no need to summarize,

Let's go back now to the given equality and place in it the results of simplifying the expressions that were detailed in A and B:

$5^3:(4^2+3^2)-(\sqrt{100}-8^2)=5^3:4^2+3^2-\sqrt{100}+8^2 \\ \downarrow\\ 59= 70\frac{13}{16}$As can be seen this equality does not hold, meaning - we got a false sentence,

So the correct answer is answer B.