/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 Evaluate the indicated quantitie... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 3

Evaluate the indicated quantities. Do not use a calculator because otherwise you will not gain the understanding that these exercises should help you attain. $$ 32^{3 / 5} $$

Short Answer

Expert verified
To evaluate the expression \(32^{3/5}\) without using a calculator, we can rewrite and simplify the expression step by step. First, rewrite the expression as a root: \(\sqrt[5]{32^3}\). Then, simplify the expression: \(\sqrt[5]{(2^5)^3}\) and apply the exponent property: \(\sqrt[5]{2^{5 \cdot 3}}\). Next, simplify the exponent: \(\sqrt[5]{2^{15}}\) and rewrite the expression as a fractional exponent: \(2^{15/5}\). Finally, simplify the fractional exponent: \(2^3\) and evaluate the expression: \(2 \cdot 2 \cdot 2 = 8\). So, \(32^{3/5} = 8\).

Step by step solution

01

Rewrite the expression as a root

To evaluate \(32^{3/5}\), we can rewrite the expression as a fractional exponent or root: \[ \sqrt[5]{32^3} \]
02

Simplify the expression

Now, we need to simplify the expression by finding \(32^3\) and then taking the 5th root of the result: \[ \sqrt[5]{(2^5)^3} \]
03

Apply the exponent property

Apply the exponent property, (a^m)^n = a^(m*n), on the expression: \[ \sqrt[5]{2^{5 \cdot 3}} \]
04

Simplify the exponent

Simplify the exponent 5 * 3: \[ \sqrt[5]{2^{15}} \]
05

Rewrite the expression as a fractional exponent

Rewrite the expression as a fractional exponent: \[ 2^{15/5} \]
06

Simplify the fractional exponent

Simplify the fractional exponent 15/5: \[ 2^3 \]
07

Evaluate the expression

Now, simply evaluate the expression \(2^3\): \[ 2 \cdot 2 \cdot 2 = 8 \] So, \(32^{3/5} = 8\).

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