Solve the Square Root Expression: Finding √9x

Question

Solve the following exercise:

9x= \sqrt{9x}=

Video Solution

Solution Steps

00:00 Simplify the following expression
00:03 The square root of a number (A) multiplied by the square root of another number (B)
00:07 Equals the square root of their product (A times B)
00:11 Apply this formula to our exercise, and convert from root 1 to two
00:17 Break down 9 to 3 squared
00:23 The square root of any number(A) squared cancels out the square
00:29 Apply this formula to our exercise
00:32 This is the solution

Step-by-Step Solution

In order to simplify the given expression, we will use two laws of exponents:

A. Definition of the root as an exponent:

an=a1n \sqrt[n]{a}=a^{\frac{1}{n}}

B. Law of exponents for dividing powers with the same base:

(ab)n=anbn (a\cdot b)^n=a^n\cdot b^n

Let's start with converting the root to an exponent using the law of exponents shown in A:

9x=(9x)12= \sqrt{9x}= \\ \downarrow\\ (9x)^{\frac{1}{2}}= Next, we will use the law of exponents shown in B and apply the exponent to each of the factors in the numerator that are in parentheses:

(9x)12=912x12=9x=3x (9x)^{\frac{1}{2}}= \\ 9^{\frac{1}{2}}\cdot x^{{\frac{1}{2}}}=\\ \sqrt{9}\sqrt{x}=\\ \boxed{3\sqrt{x}} In the last steps, we will multiply the half exponent by each of the factors in the numerator, returning to the root form, that is, according to the definition of the root as an exponent shown in A (in the opposite direction) and then we will calculate the known fourth root result of the number 9.

Therefore, the correct answer is answer D.

Answer

3x 3\sqrt{x}