1. $x$ must be greater than $0$
2. We want to find the value of $(√{36x}+√{16x})^2$

Develop a Plan

The question has an expression with radical signs and an exponent, but the answer choices have neither. So let’s simplify our expression to get rid of the radical signs and the exponent. Let’s keep in mind our Order of Operations (PEMDAS:
Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) while simplifying our expression.

Solve the Question

Using PEMDAS, we first look inside the Parentheses. Here we see two terms added together, and they both have radical signs. We know from our knowledge of Exponents and Roots that we can only add together radical terms if they have the same, exact thing underneath their radicals. So let’s try to simplify each of these terms first then see if we can add them together.

From our Exponents and Roots math skill, we know that we can remove something from underneath a square root sign if we can write it as a number squared. As an example, if we have $√25$, we can rewrite it as $√5^2$, and then remove the $5$ from underneath the radical sign, giving us $√25=√5^2=5$. Let’s use this technique to simplify each radical term inside the parentheses now.

$$√{4x}=√{2^2·x}=2√x$$
$$√{9x}=√{3^2·x}=3√x$$

Nicely done! Let’s put these two terms inside of our parentheses now.

$$(√{4x}+√{9x})^2=(2√x+3√x)^2$$

We know from our Exponents and Roots math skill that we can add radicals together if the number or expression underneath the radical is the exact same. For example, $3√5+4√5=(3+4)√5=7√5$. It’s kind of like saying $3$ apples plus $4$ apples gives us $7$ apples, except we just replace “apples” with “square root of $5$” in this example.

The two terms inside our parentheses have the exact same thing underneath the radical sign $(x)$, so let’s add them together.

$$(2√x+3√x)^2=([2+3]√x)=(5√x)^2$$

Our expression definitely looks a lot cleaner now! Still need to rewrite it to get rid of the radical and the exponent though.

We know from our Exponents and Roots math skill that we can distribute exponents from outside parentheses inside the parentheses by using the Power Rule of Exponents, which tells us that the exponent being distributed inside the parentheses MUST be distributed to each individual term that is multiplied or divided within the parentheses. So let’s distribute the $2$ exponent inside the parentheses now.

$$(5√x)^2=(5·√x)^2 = 5^2·(√x)^2$$

We know that $5^2=25$, so let’s put that in to get rid of the $2$ exponent modifying the $5$.

$$5^2·(√x)^2=25·(√x)^2$$

We remember seeing in our Exponents and Roots math skill that squaring a square root cancels out both the radical sign and the exponent. In math terms, we call them inverse operations since they accomplish the opposite, or inverse, task of each other. This is similar to how addition is the inverse operation of subtraction. Since the radical sign and the square root cancel each other out, we get:

$$25·(√x)^2=25·x=25x$$

The correct answer is D, $25x$.

What Did We Learn

This question was great practice combining our Exponents and Roots math skill with our knowledge of the Order of Operations (PEMDAS).

Another takeaway from this question is that we can add or subtract radicals ONLY if they have the exact same thing underneath the radical sign. So while we could not directly add together $(√{4x}+√{9x})$ since they have different terms underneath the square root sign, because $4x$ is not the exact same thing as $9x$. However, we were able to add together $(2√x+3√x)$ since they both have the exact same thing underneath the square root sign $(√x)$, giving us: $2√x+3√x=5√x$.

 

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