Compare Complex Expressions: Evaluating 3³-5² with Square Roots and Exponents

To solve the given problem and determine whether it is an equality or inequality, we need to simplify the expressions on either side.

We can deal with each of them separately and simplify them, however, a more efficient way of working will be to deal with the more complex parts of these expressions separately, that is, with the expressions containing roots,

It's important to emphasize that in generalwe want to learn to solve without using a calculator by using our algebraic tools and the laws of exponents. Let's begin:

a. Let's start with the first part of the expression on the right:

$\sqrt{10}+\sqrt{2}\cdot\sqrt{5}$(We'll focus on the expression inside the parentheses first and then continue outwards).

Let's recall two laws of exponents:

a.1: Defining a root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

a.2: The law for an exponent applied to parentheses containing multiplication, but in the opposite direction:

$x^n\cdot y^n= (x\cdot y)^n$

Usually we would replace roots with exponents, but for now we won't do that. In the meantime just understand that according to the law of defining a root as an exponent mentioned in a.1, the root is actually an exponent and therefore all laws of exponents apply to it, especially the law of exponents mentioned ina.2. Let's apply this understanding to the expression in question:

$\sqrt{10}+\sqrt{2}\cdot\sqrt{5} = \sqrt{10}+\sqrt{2\cdot5}= \sqrt{10}+\sqrt{10}$

In the first step, we notice that the second term (i.e., the product of roots) is actually a product between two terms raised to the same exponent (which is half the power of the square root). Therefore using to the law of exponents mentioned in a.2, we can multiply the bases of the terms under the same square root, and in the next stage we simplify the expression in the root.

From here we note that this expression can be factored by a common factor:

$\sqrt{10}+\sqrt{10}=(1+1)\sqrt{10}=2\cdot\sqrt{10}$

We used the commutative property of multiplication, that is, the common factor we took out:$\sqrt{10}$We took out to the right of the parentheses (instead of to their left),

Let's continue to the second problematic expression on the right:

$\sqrt{48}\cdot\sqrt{3}$And we'll deal with it the same way as in the previous part:

$\sqrt{48}\cdot\sqrt{3} = \sqrt{48\cdot3}=\sqrt{144}$Again, in the first step, we notice that the expression (i.e., the product of roots) is actually a product between two terms raised to the same exponent (which is half the power of the square root), therefore according to the law of exponents mentioned in a.2, we can multiply the bases of the terms under the same exponent, and in the next step we simplify the expression in the root,

From here we'll return to the original expression in the problem (i.e., the expression on the right) and calculate it in full, using the simplified expressions we got above:

$4^2(\sqrt{10}+\sqrt{2}\cdot\sqrt{5})^2+\sqrt{48}\cdot\sqrt{3} = \\ 4^2(2\cdot\sqrt{10})^2+\sqrt{144}$We just substitute what we calculated earlier in the two parts above - in place of the expression in parentheses and in place of the second term.

INext, let's recall again the law of exponents mentioned above in a.2, that is, the law for an exponent applied to parentheses containing a product but in the normal way.

Let's apply this law to the expression we got in the last stage:

$4^2(2\cdot\sqrt{10})^2+\sqrt{144} =4^2\cdot2^2\cdot(\sqrt{10})^2+\sqrt{144}=\\ 16\cdot4\cdot10+12$We apply the law of exponents mentioned in a.2 above and applied the exponent to each of the multiplication terms in the parentheses.

Next, we apply the exponent to the square root while remembering that these are actually two inverse operations and therefore cancel each other out and simultaneously simplify the other terms by applying the square root to the second term on the left, and the exponents to the first term.

Let's finish solving. We got that the expression is:

$16\cdot4\cdot10+12 =640+12=652$Let's summarize this part:

We got that the expression on the right is:

$4^2(\sqrt{10}+\sqrt{2}\cdot\sqrt{5})^2+\sqrt{48}\cdot\sqrt{3}=\\ 4^2(2\cdot\sqrt{10})^2+\sqrt{144} =16\cdot4\cdot10+12 =\\ 652$b. Let's continue to the expression on the left and start by writing it in the standard fraction notation while keeping in mind the order of operations, that is - parentheses, exponents, multiplication/ division, addition/ subtraction.

$3^3-5^2:(2^2+1)= 3^3-\frac{5^2}{2^2+1}$Since the division here refers to the entire expression in parentheses, we inserted it in its entirety into the denominator of the fraction,

Let's simplify this expression. We'll focus on the fraction, but first let's recall the law of multiplying exponents with identical bases:

b.1:

$\frac{a^m}{a^n}=a^{m-n}$Let's simplify the expression. We'll start by simplifying the numerator of the fraction and continue by applying the law of exponents mentioned above:

$3^3-\frac{5^2}{2^2+1} =3^3-\frac{5^2}{4+1} =3^3-\frac{5^2}{5} =\\ 3^3-5^{2-1}=3^3-5$In the first and second parts, we simplified the numerator of the fraction, in the third part we applied the law of exponents mentioned in b.1 and in the following parts we simplified the resulting expression.

$$Let's finish the calculation:

$3^3-5 =27-5=22$And to summarize:

$3^3-5^2:(2^2+1)= 3^3-\frac{5^2}{2^2+1} =\\ 3^3-\frac{5^2}{5} = 3^3-5^{2-1}=3^3-5=22$

Now let's return to the original problem and replace what we got for a' and b':

$3^3-5^2:(2^2+1)_{\textcolor{red}{_{——}}\text{ }}4_{\text{\textcolor{red}{ }}}^2(\sqrt{10}+\sqrt{2}\cdot\sqrt{5})^2+\sqrt{48}\cdot\sqrt{3} \\ \downarrow\\ 22\text{ }\text{\textcolor{red}{\_\_}}\text{ }652$Therefore it's clear that this is not an equality but an inequality and that the expression on the left is smaller than the expression on the right,meaning that:

22\text{ }\text{\textcolor{red}{<}}\text{ }652 Therefore the correct answer is answer a'.