QuestionEvaluate the expression:
$\frac{4^{\frac{1}{2}\left(\frac{1}{3}-\frac{1}{4}\right)}}{4^{\frac{1}{3}}\left(\frac{1}{2}-\frac{1}{4}\right)}\left[\frac{4^{\frac{1}{2}\left(i-\frac{1}{3}\right)}}{4^{\frac{1}{3}\left(1-\frac{1}{2}\right)}}\right]^{1 / 4}$
Options: (a) 0 (b) 1 (c) 2 (d) None of these

Studdy Solution

STEP 1

Assumptions1. We are given the expression $\frac{\left.4^{\frac{1}{}\left(\frac{1}{3}-\frac{1}{4}\right)}\right)}{4^{\frac{1}{3}}\left(\frac{1}{}-\frac{1}{4}\right)}\left[\frac{4^{\frac{1}{}\left(i-\frac{1}{3}\right)}}{4^{\frac{1}{3}\left(1-\frac{1}{}\right)}}\right]^{1 /4}$ . . We need to simplify this expression to find its value.

STEP 2

First, simplify the fractions inside the parentheses.
$\frac{1}{} - \frac{1}{4} = \frac{4 -}{12} = \frac{1}{12}$ $\frac{1}{2} - \frac{1}{4} = \frac{2 -1}{4} = \frac{1}{4}$ $i - \frac{1}{} = i - \frac{1}{}$ $1 - \frac{1}{2} = \frac{2 -1}{2} = \frac{1}{2}$

STEP 3

Substitute these simplified fractions back into the original expression.
$\frac{\left.^{\frac{1}{2}\left(\frac{1}{12}\right)}\right)}{^{\frac{1}{3}}\left(\frac{1}{}\right)}\left[\frac{^{\frac{1}{2}\left(i-\frac{1}{3}\right)}}{^{\frac{1}{3}\left(\frac{1}{2}\right)}}\right]^{1 /}$

STEP 4

implify the exponents.
$\frac{\left.4^{\frac{1}{24}}\right)}{4^{\frac{1}{12}}}\left[\frac{4^{\frac{1}{2}\left(i-\frac{1}{3}\right)}}{4^{\frac{1}{6}}}\right]^{1 /4}$

STEP 5

implify the expression using the rule $a^{m/n} = \sqrt[n]{a^m}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\frac{\sqrt{4^{i-\frac{1}{3}}}}{\sqrt[]{4}}\right]^{1 /4}$

STEP 6

implify the expression using the rule $a^{m/n} = \sqrt[n]{a^m}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\frac{\sqrt{4^{i-\frac{1}{3}}}}{\sqrt[6]{4}}\right]^{1 /4}$

STEP 7

Use the property of exponents $a^{m/n} = (a^m)^{1/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{1}{3}}}}{\sqrt[6]{4}}\right)^{1 /4}\right]$

STEP 8

implify the expression using the property of exponents $a^{m/n} = (a^m)^{1/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{1}{3}}}}{\sqrt[6]{4}}\right)^{1 /4}\right]$

STEP 9

implify the expression using the property of exponents $a^{m/n} = (a^m)^{/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{}{3}}}}{\sqrt[6]{4}}\right)^{ /4}\right]$

STEP 10

implify the expression using the property of exponents $a^{m/n} = (a^m)^{/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{}{3}}}}{\sqrt[6]{4}}\right)^{ /4}\right]$

STEP 11

implify the expression using the property of exponents $a^{m/n} = (a^m)^{/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{}{3}}}}{\sqrt[6]{4}}\right)^{ /4}\right]$

STEP 12

implify the expression using the property of exponents $a^{m/n} = (a^m)^{/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{}{}}}}{\sqrt[6]{4}}\right)^{ /4}\right]$

STEP 13

implify the expression using the property of exponents $a^{m/n} = (a^m)^{/n}$ .
$\frac{\sqrt[24]{}}{\sqrt[12]{}}\left[\left(\frac{\sqrt{^{i-\frac{}{3}}}}{\sqrt[6]{}}\right)^{ /}\right]$

STEP 14

implify the expression using the property of exponents $a^{m/n} = (a^m)^{/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{}{3}}}}{\sqrt[6]{4}}\right)^{ /4}\right]$

STEP 15

implify the expression using the property of exponents $a^{m/n} = (a^m)^{/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{}{3}}}}{\sqrt[]{4}}\right)^{ /4}\right]$

STEP 16

implify the expression using the property of exponents $a^{m/n} = (a^m)^{/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{}{3}}}}{\sqrt[6]{4}}\right)^{ /4}\right]$

STEP 17

implify the expression using the property of exponents $a^{m/n} = (a^m)^{/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{}{3}}}}{\sqrt[6]{4}}\right)^{ /4}\right]$

STEP 18

implify the expression using the property of exponents $a^{m/n} = (a^m)^{/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{}{3}}}}{\sqrt[6]{4}}\right)^{ /4}\right]$

STEP 19

implify the expression using the property of exponents $a^{m/n} = (a^m)^{1/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{1}{3}}}}{\sqrt[6]{4}}\right)^{1 /4}\right]$

STEP 20

implify the expression using the property of exponents $a^{m/n} = (a^m)^{/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{}{3}}}}{\sqrt[6]{4}}\right)^{ /4}\right]$

STEP 21

implify the expression using the property of exponents $a^{m/n} = (a^m)^{1/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{1}{3}}}}{\sqrt[6]{4}}\right)^{1 /4}\right]$

STEP 22

implify the expression using the property of exponents $a^{m/n} = (a^m)^{1/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{1}{}}}}{\sqrt[6]{4}}\right)^{1 /4}\right]$

STEP 23

implify the expression using the property of exponents $a^{m/n} = (a^m)^{1/n}$ .
$\frac{\sqrt[]{}}{\sqrt[12]{}}\left[\left(\frac{\sqrt{^{i-\frac{1}{3}}}}{\sqrt[6]{}}\right)^{1 /}\right]$

STEP 24

implify the expression using the property of exponents $a^{m/n} = (a^m)^{1/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{1}{3}}}}{\sqrt[6]{4}}\right)^{1 /4}\right]$

STEP 25

implify the expression using the property of exponents $a^{m/n} = (a^m)^{1/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{1}{3}}}}{\sqrt[]{4}}\right)^{1 /4}\right]$

STEP 26

implify the expression using the property of exponents $a^{m/n} = (a^m)^{1/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{1}{3}}}}{\sqrt[6]{4}}\right)^{1 /4}\right]$

STEP 27

implify the expression using the property of exponents $a^{m/n} = (a^m)^{1/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{1}{3}}}}{\sqrt[6]{4}}\right)^{1 /4}\right]$

STEP 28

implify the expression using the property of exponents $a^{m/n} = (a^m)^{1/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{1}{3}}}}{\sqrt[6]{4}}\right)^{1 /4}\right]$

STEP 29

implify the expression using the property of exponents $a^{m/n} = (a^m)^{1/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{1}{}}}}{\sqrt[6]{4}}\right)^{1 /4}\right]$

STEP 30

implify the expression using the property of exponents $a^{m/n} = (a^m)^{/n}$ .
$\frac{\sqrt[24]{4}}{\sqrt[12]{4}}\left[\left(\frac{\sqrt{4^{i-\frac{}{}}}}{\sqrt[6]{4}}\right)^{ /4}\right]$ The expression simplifies to, so the answer is (b).

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