Solve the Square Root Expression: Simplifying √(100x²)

Question

Solve the following exercise:

100x2= \sqrt{100x^2}=

Video Solution

Solution Steps

00:06 Let's simplify this expression step by step.
00:10 Take the square root of A, and multiply it by the square root of B.
00:15 This equals the square root of the product, A times B.
00:19 Now, let's use this formula to solve our exercise. Break it down into two roots.
00:25 Break down one hundred into ten squared.
00:30 Remember, the square root of A squared cancels out the square.
00:36 Let's use this rule in our exercise.
00:39 And there you have it! That's the solution.

Step-by-Step Solution

In order to simplify the given expression, apply the following three laws of exponents:

a. Definition of root as an exponent:

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

b. Law of exponents for an exponent applied to terms in parentheses:

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

c. Law of exponents for an exponent raised to an exponent:

(am)n=amn (a^m)^n=a^{m\cdot n}

Begin by converting the fourth root to an exponent using the law of exponents mentioned in a.:

100x2=(100x2)12= \sqrt{100x^2}= \\ \downarrow\\ (100x^2)^{\frac{1}{2}}=

We'll continue, using the law of exponents mentioned in b. and apply the exponent to each factor in the parentheses:

(100x2)12=10012(x2)12 (100x^2)^{\frac{1}{2}}= \\ 100^{\frac{1}{2}}\cdot(x^2)^{{\frac{1}{2}}}

We'll continue, once again using the law of exponents mentioned in c. and perform the exponent applied to the term with an exponent in parentheses (the second factor in the multiplication):

10012(x2)12=10012x212=10012x1=100x=10x 100^{\frac{1}{2}}\cdot(x^2)^{{\frac{1}{2}}} = \\ 100^{\frac{1}{2}}\cdot x^{2\cdot\frac{1}{2}}=\\ 100^{\frac{1}{2}}\cdot x^{1}=\\ \sqrt{100}\cdot x=\\ \boxed{10x}

In the final steps, we first converted the power of one-half applied to the first factor in the multiplication back to the fourth root form, again, according to the definition of root as an exponent mentioned in a. (in reverse) and then calculated the known fourth root of 100.

Therefore, the correct answer is answer d.

Answer

10x 10x