Powers and Roots: Using parentheses

To solve the given power expression, we need to apply the formula for powers of a fraction. The expression we are given is:
$\left(\frac{2}{3}\right)^3$

Let's break down the steps:

• When we raise a fraction to a power, we apply the exponent to both the numerator and the denominator separately. This means raising both 2 and 3 to the power of 3.
• Thus, we calculate:
  $2^3 = 8$ and $3^3 = 27$.
• Therefore, $\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}$.

So, the result of the expression $\left(\frac{2}{3}\right)^3$ is $\frac{8}{27}$.

Let's solve this problem step by step using the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction):

1. First, let's focus on what's inside the parentheses: $\sqrt{36}+9$

2. We need to evaluate the square root first:

• $\sqrt{36} = 6$ (because $6 \times 6 = 36$)

3. Now our expression looks like this: $2\times(6+9)$

4. Next, we perform the addition inside the parentheses:

• $6 + 9 = 15$

5. Our expression is now: $2\times15$

6. Finally, we perform the multiplication:

• $2 \times 15 = 30$

Therefore, $2\times(\sqrt{36}+9) = 30$

This matches the provided correct answer of 30.

Let's solve the expression $(2+1\times2)^2$ step-by-step, adhering to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Firstly, handle the expression inside the parentheses $(2+1\times2)$:

• Within the parentheses, according to PEMDAS, we first perform the multiplication $1\times2$ which equals $2$.
• Now, the expression inside the parentheses becomes $(2+2)$.
• Next, perform the addition: $2+2=4$.

Now the expression simplifies to $4^2$.

Second, handle the exponent:

• Calculate the square of 4: $4^2 = 16$.

Thus, the final answer is $16$.

Remember that according to the order of arithmetic operations, exponents precede multiplication and division, which precede addition and subtraction (and parentheses always precede everything).

So first calculate the values of the terms with exponents and then subtract the results:

$(5-2)^2-2^3 =3^2-2^3=9-8=1$Therefore, the correct answer is option C.

Previously mentioned in the order of operations that exponents come before multiplication and division which come before addition and subtraction (and parentheses always come before everything),

Let's calculate then first the value of the expression inside the parentheses (by calculating the roots inside the parentheses first) :

$(\sqrt{9}-\sqrt{4})^2\cdot4^2-5^1 =(3-2)^2\cdot4^2-5^1 =1^2\cdot4^2-5^1$where in the second stage we simplified the expression in parentheses,

Next we'll calculate the values of the terms with exponents:

$1^2\cdot4^2-5^1 =1\cdot16-5$then we'll calculate the results of the multiplications

$1\cdot16-5 =16-5$and after that we'll perform the subtraction:

$16-5=11$Therefore the correct answer is answer B.

Previously mentioned in the order of operations that exponents come before multiplication and division which come before addition and subtraction (and parentheses always come before everything),

Let's calculate first the value of the expression inside the parentheses (by calculating the values of the terms inside the parentheses first) :

$(10^2-2\cdot5):3^2 = (100-10):3^2 =90:3^2=\frac{90}{3^2}$where in the second stage we simplified the expression in parentheses, and in the next stage we wrote the division operation as a fraction,

Next we'll calculate the value of the term in the fraction's numerator by performing the exponent, and in the next stage we'll perform the division (essentially reducing the fraction):

$\frac{90}{3^2} =\frac{\not{90}}{\not{9}}=10$Therefore the correct answer is answer D.

Previously mentioned in the order of arithmetic operations, exponents come before multiplication and division which come before addition and subtraction (and parentheses always come before everything),

Let's calculate first the value of the expression inside the parentheses (by calculating the values of the root terms inside the parentheses first) :

$(\sqrt{100}-\sqrt{9})^2:7 = (10-3)^2:7 =7^2:7=\frac{7^2}{7}$where in the second step we simplified the expression in parentheses, and in the next step we wrote the division operation as a fraction,

Next we'll calculate the value of the numerator by performing the exponentiation, and in the next step we'll perform the division (essentially reducing the fraction):

$\frac{7^2}{7} =\frac{\not{49}}{\not{7}}=7$Therefore the correct answer is answer A.

Previously mentioned in the order of arithmetic operations, exponents come before multiplication and division which come before addition and subtraction (and parentheses always come before everything),

Let's calculate first the value of the expression inside the parentheses (by calculating the values of the terms with exponents inside the parentheses first) :

$5:(13^2-12^2) =5:(169-144) =5:25=\frac{5}{25}$where in the second step we simplified the expression in parentheses, and in the next step we wrote the division operation as a fraction,

Then we'll perform the division (we'll actually reduce the fraction):

$\frac{\not{5}}{\not{25}}=\frac{1}{5}$Therefore the correct answer is answer C.

To solve the expression $3-(5^2:5)^2+7^2$, we should follow the order of operations, which is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Here are the steps to solve the expression:

1. Evaluate the exponents

• Calculate $5^2$ which equals $25$.

• Calculate $7^2$ which equals $49$.

2. Evaluate expressions inside parentheses

• The expression inside the parentheses is $5^2:5$ which simplifies to $25:5 = 5$.

3. Evaluate the expression inside the parentheses raised to a power

• The simplified expression now is $(5)^2$, which is $25$.

4. Substitute back into the expression

• The original expression now becomes: $3 - 25 + 49$.

5. Perform the addition and subtraction from left to right

• First, calculate $3 - 25$ which equals $-22$.

• Then, $-22 + 49$ equals $27$.

Therefore, the final result of the expression $3-(5^2:5)^2+7^2$ is $27$.

Let's walk through the steps to solve the expression $(4^2:8):2+3^2$ using the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

• First, resolve the expression inside the parentheses: $4^2:8$

  • The exponent comes first:

    $4^2 = 16$, so the expression now is $16:8$.

• Next, perform the division inside the parentheses: $16:8$ equals 2. So the expression within the parentheses simplifies to 2.

• Now, we replace the original expression with this simplified result:

  $2:2+3^2$

• We perform the division: $2:2 = 1$.

• Substitute back into the expression:

  $1+3^2$

• Next, calculate the exponent:

  $3^2 = 9$.

• Finally, add the results:

  $1 + 9 = 10$.

Thus, the solution to the expression $(4^2:8):2+3^2$ is 10.

To solve the expression $(18-10)^2+3^3$, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

• Step 1: Parentheses
  First, solve the expression inside the parentheses: $18-10$.
  $18-10 = 8$

• Step 2: Exponents
  Next, apply the exponents to the numbers:
  $(8)^2$ and $3^3$.
  $8^2 = 64$
  $3^3 = 27$

• Step 3: Addition
  Finally, add the results of the exponentiations:
  $64 + 27$
  $64 + 27 = 91$

Thus, the final answer is $91$.

Previously mentioned in the order of operations that exponents come before multiplication and division which come before addition and subtraction (and parentheses always come before everything),

Let's calculate first the value of the expression inside the parentheses (by calculating the values of the terms with exponents inside the parentheses first) :

$(4^2+3^2):\sqrt{25} =(16+9):\sqrt{25} =25:\sqrt{25} =\frac{25}{\sqrt{25}}$where in the second step we simplified the expression in parentheses, and in the next step we wrote the division as a fraction,

$$we'll continue and calculate the value of the square root in the denominator:

$\frac{25}{\sqrt{25}} =\frac{25}{5}$and then we'll perform the division (reducing the fraction essentially):

$\frac{25}{5} =5$Therefore the correct answer is answer B.

Previously mentioned in the order of operations that exponents come before multiplication and division which come before addition and subtraction (and parentheses always come before everything),

Let's calculate first the value of the expression inside the parentheses (by calculating the values of the terms with exponents and roots inside the parentheses first) :$(\sqrt{25}-2^2)^3+2^3= (5-4)^3+2^3=1^3+2^3$where in the second stage we simplified the expression in parentheses,

Next we'll calculate the values of the terms with exponents and perform the addition operation:

$1^3+2^3=1+8=9$Therefore the correct answer is answer A.

Previously mentioned in the order of arithmetic operations, exponents come before multiplication and division which come before addition and subtraction (and parentheses always come before everything),

We will therefore first calculate the value of the expression inside the parentheses in the leftmost term (by calculating the values of the terms with exponents and roots inside the parentheses first) :$7:(5^2-\sqrt{16})\cdot3+\sqrt{3}\cdot\sqrt{3} = 7:(25-4)\cdot3+\sqrt{3}\cdot\sqrt{3}=7:21\cdot3+\sqrt{3}\cdot\sqrt{3}$where in the second stage we simplified the expression in parentheses,

Then for convenience we'll write the division operation in the leftmost term, where:

$7:21\cdot3+\sqrt{3}\cdot\sqrt{3} =\frac{7}{21}\cdot3+\sqrt{3}\cdot\sqrt{3}$Let's pause here for a moment and notice that we can actually write the expression in the following way:

$7\cdot\frac{1}{21}\cdot3+\sqrt{3}\cdot\sqrt{3}$And using the commutative property of multiplication we can write it as:

$7\cdot\frac{1}{21}\cdot3+\sqrt{3}\cdot\sqrt{3} = 7\cdot3\cdot\frac{1}{21}+\sqrt{3}\cdot\sqrt{3}$In other words, we can switch the order of operations - multiplying by 3 and dividing by 21 (which is represented by the fraction). If we return to writing it using division operation, it will look like this:

$7\cdot3\cdot\frac{1}{21}+\sqrt{3}\cdot\sqrt{3} =7\cdot3:21+\sqrt{3}\cdot\sqrt{3}$We could have done this at the beginning of the solution, but we must ensure that the operation (division or multiplication) before the number is well defined and that division operation can never be written first. Note that the commutative property is not well defined for division and therefore cannot be used for division, but after converting to a fraction (which is actually being multiplied) the commutative property is well defined (since it's now multiplication),

Let's return to the problem, in the last stage we got:

$\frac{7}{21}\cdot3+\sqrt{3}\cdot\sqrt{3}$ We'll continue and perform both multiplication and division in the first term, and calculate the value of the second term from the left by performing the multiplication:

$\frac{7}{21}\cdot3+\sqrt{3}\cdot\sqrt{3} = \frac{7\cdot3}{21}+(\sqrt{3})^2= \frac{21}{21}+3=1+3=4$When in the first stage we remembered that multiplication by a fraction is actually multiplication by the numerator of the fraction for the first term from the left, and for the second term from the left we remembered that multiplying a number by itself is raising the number to the power of 2. In the following stages we calculated the result of multiplication by the fraction's numerator and also remembered the definition of root for the second term (which states that root and exponent of the same order are inverse operations). Finally, we reduced the fraction (actually performed the division operation) and calculated the result of the addition operation

Therefore the correct answer is answer D.

The given expression is: $(3^2+2^2)^2:(\sqrt{256}-\sqrt{9})-\sqrt{9}\cdot\sqrt{9}$

Let's solve it step by step:

Step 1: Solve inside the parentheses

Calculate $3^2$ and $2^2$:

• $3^2 = 9$
• $2^2 = 4$

Add the results: $9 + 4 = 13$

Now, the expression becomes $(13)^2 : (\sqrt{256} - \sqrt{9}) - \sqrt{9}\cdot\sqrt{9}$

Step 2: Calculate the powers and roots

• $(13)^2 = 169$
• $\sqrt{256} = 16$
• $\sqrt{9} = 3$

Substitute these back into the expression: $169 : (16 - 3) - 3\cdot3$

Step 3: Simplify the expression

• The expression in the parentheses: $16 - 3 = 13$

Now the expression is: $169 : 13 - 9$

Step 4: Perform division

Divide $169$ by $13$:

• $169 : 13 = 13$

Now the expression is: $13 - 9$

Step 5: Perform subtraction

Subtract $9$ from $13$:

• $13 - 9 = 4$

The final answer is $4$.

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

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

$(5^2+3):2^2$Let's simplify this expression while remembering the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and parentheses come before everything else, therefore we'll start by simplifying the expression inside the parentheses, this is done by calculating the numerical value of the terms with exponents within them, then we'll calculate the addition operation in the parentheses:

$(5^2+3):2^2 =\\ (25+3):2^2 =\\ 28:2^2$We'll continue and remember that exponents come before division, therefore, we'll first calculate the term with the exponent which is the divisor in the expression (in fact, if we were to convert the division operation to a fraction, this term would be in the denominator), then we'll calculate the result of the division operation:

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

$(5^2+3):2^2 =\\ 28:2^2= \\ 28:4 =\\ 7$B. Let's continue with 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 adhering to the order of operations mentioned earlier, therefore, we'll again start by simplifying the expression inside the parentheses, this is first done by calculating the numerical value of the term with the exponent (since exponents come before division), then we'll perform the division operation on the second term from the left (in parentheses), simultaneously we'll calculate the numerical value of the term with the exponent (the first from the left) and then we'll perform the addition operation:

$5^2+(3:2^2) =\\ 5^2+(3:4)=\\ 25+\frac{3}{4}=\\ 25\frac{3}{4}$Note that since the division operation yielded a non-whole number we settled for converting this operation to a fraction, finally we performed the addition operation between the whole number and the fraction and wrote the result as a mixed number, this fraction can be converted to a decimal but there's no need for that,

Note that in this expression the parentheses are actually meaningless because multiplication and division come before addition and subtraction anyway, but good practice says that if they're noted in the problem, they should be given precedence in the approach,

We've finished simplifying the expression on the right side of the equation, since the calculation is short there's no need to summarize,

Let's return then to the original equation and substitute in place of the expressions on both sides the results of the simplifications detailed in A and B in order to determine its correctness (or incorrectness):

$(5^2+3):2^2=5^2+(3:2^2) \\ \downarrow\\ 7=25\frac{3}{4}$Now we can definitively determine that the given equation is incorrect, meaning - we have a false statement,

Therefore the correct answer is answer B.