Simplify the Square Root Expression: √(12x⁴)

Question

Solve the following exercise:

12x4= \sqrt{12x^4}=

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)
00:10 Apply this formula to our exercise, and convert from root 1 to two
00:19 Break down X to the fourth into X squared squared
00:26 The square root of any number (A) squared cancels out the square
00:29 Apply this formula to our exercise
00:33 This is the solution

Step-by-Step Solution

In order to simplify the given expression, we will 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.:

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

Next, apply the law of exponents mentioned in b. and proceed to apply the exponent to each factor inside of the parentheses:

(12x4)12=1212(x4)12 (12x^4)^{\frac{1}{2}}= \\ 12^{\frac{1}{2}}\cdot(x^4)^{{\frac{1}{2}}}

Let's continue, using the law of exponents mentioned in c. and perform the exponent operation on the term with an exponent in the parentheses (the second term in the product):

1212(x4)12=1212x412=1212x2=12x2= 12^{\frac{1}{2}}\cdot(x^4)^{{\frac{1}{2}}} = \\ 12^{\frac{1}{2}}\cdot x^{4\cdot\frac{1}{2}}=\\ 12^{\frac{1}{2}}\cdot x^{2}=\\ \boxed{\sqrt{12} x^2}=\\ In the final step, we converted the one-half exponent on the first term in the product back to a fourth root form, again, according to the definition of root as an exponent mentioned in a. (in the reverse direction).

Therefore, the correct answer is answer c.

Answer

12x2 \sqrt{12}x^2