Solve the Square Root Expression: Simplifying √(25x⁴)

Question

Solve the following exercise:

25x4= \sqrt{25x^4}=

Video Solution

Solution Steps

00:00 Simplify the following expression
00:03 The square root of a number (A) multiplied by square root of another number (B)
00:07 equals the square root of their product (A times B)
00:10 We will apply this formula to our exercise, and convert root 1 into two
00:16 Break down 25 to 5 squared
00:20 Break down X to the fourth power into X squared squared
00:27 The square root of any number(A) squared cancels out the square
00:30 Apply this formula to our exercise
00:34 This is 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.:

25x4=(25x4)12= \sqrt{25x^4}= \\ \downarrow\\ (25x^4)^{\frac{1}{2}}=

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

(25x4)12=2512(x4)12 (25x^4)^{\frac{1}{2}}= \\ 25^{\frac{1}{2}}\cdot(x^4)^{{\frac{1}{2}}}

We'll continue, 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):

2512(x4)12=2512x412=2512x2=25x2=5x2 25^{\frac{1}{2}}\cdot(x^4)^{{\frac{1}{2}}} = \\ 25^{\frac{1}{2}}\cdot x^{4\cdot\frac{1}{2}}=\\ 25^{\frac{1}{2}}\cdot x^{2}=\\ \sqrt{25}\cdot x^2=\\ \boxed{5x^2}

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 the reverse direction) and then calculated the known fourth root of 25.

Therefore, the correct answer is answer a.

Answer

5x2 5x^2