Solve Fourth Root Division: Simplifying (Fourth Root of 128)/(Fourth Root of 8)

Question

Solve the following exercise:

128484= \frac{\sqrt[4]{128}}{\sqrt[4]{8}}=

Video Solution

Solution Steps

00:00 Solve the following problem
00:03 The root of number (A) divided by root of number (B)
00:07 Is the same as the root of fraction (A divided by B)
00:15 Apply this formula to our exercise
00:25 Calculate 128 divided by 8
00:34 Break down 4 into 2 x 2
00:41 When there is a root with a power that is a multiple
00:44 We can divide it into a root of a power
00:47 Within the root of the second power
00:50 We apply this formula to our exercise
00:57 The root of the power 2 is a "regular" root
01:02 Break down 16 to 4 squared
01:06 The root of any squared number cancels out the square
01:14 Break down 4 to 2 squared
01:17 This is the solution

Step-by-Step Solution

Introduction:

We will address the following two laws of exponents:

a. The definition of root as an exponent:

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

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

(ab)n=anbn (\frac{a}{b})^n=\frac{a^n}{b^n}

Note:

By combining these two laws of exponents mentioned in a (in the first and third steps) and b (in the second step ), we can derive another new rule:

abn=(ab)1n=a1nb1n=anbnabn=anbn \sqrt[n]{\frac{a}{b}}=\\ (\frac{a}{b})^{\frac{1}{n}}=\\ \frac{a^{\frac{1}{n}}}{ b^{\frac{1}{n}}}=\\ \frac{\sqrt[n]{a}}{ \sqrt[n]{ b}}\\ \downarrow\\ \boxed{ \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{ \sqrt[n]{ b}}}

Therefore, in solving the problem, meaning - simplifying the given expression, we will apply the new rule studied in the introduction:

abn=anbn \boxed{ \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{ \sqrt[n]{ b}}}

We'll start by simplifying the expression using the rule we studied in the introduction (however this time in the opposite direction, meaning we'll insert the product of roots as a product of terms under the same root) We'll then proceed to perform the multiplication under the root and finally we'll perform the fifth root operation:

128484=12884=164=2 \frac{\sqrt[4]{128}}{\sqrt[4]{8}}= \\ \sqrt[4]{\frac{128}{8}}=\\ \sqrt[4]{16}=\\ \boxed{2}

Therefore, the correct answer is answer B.

Answer

2