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Chapter 3: Problem 2

Evaluate the indicated quantities. Do not use a calculator because otherwise you will not gain the understanding that these exercises should help you attain. $$ 8^{5 / 3} $$

Short Answer

Expert verified
The short answer is to first rewrite \(8^{5/3}\) as a radical expression, then simplify and evaluate it step by step: \[\begin{aligned} 8^{5/3} &= \sqrt[3]{8^5} \\ &= \sqrt[3]{(2^3)^5} \\ &= \sqrt[3]{2^{15}} \\ &= 2^{15/3} \\ &= 2^5 \\ &= 32 \end{aligned}\] So, \(8^{5/3} = 32\).

Step by step solution

01

Rewrite the expression as a radical

To begin, we rewrite the expression \(8^{5/3}\) as a radical expression. Recall that for any positive real number a and any positive integers m and n, \[a^{m/n} =\sqrt[n]{a^m}\] Using this rule, we can rewrite the expression as follows: \[(8^{5/3}) = \sqrt[3]{8^5}\]
02

Simplify the expression

We know that \(8 = 2^3\). We'll substitute this into the expression, then use the exponent rule \((a^m)^n = a^{(m\cdot n)}\) to simplify further: \[\sqrt[3]{(2^3)^5} = \sqrt[3]{2^{(3 \cdot 5)}} = \sqrt[3]{2^{15}}\]
03

Evaluate the cube root

Since we are taking a cube root, we want to see if there is an exponent in the radicand that is divisible by 3. 15 is divisible by 3 (15 = 3*5), so we can take the cube root: \[\sqrt[3]{2^{15}} = 2^{15/3} = 2^5\]
04

Evaluate the power

Finally, calculate \(2^5\): \[2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32\] The evaluated expression is: \[8^{5/3} = 32\]

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Most popular questions from this chapter

Evaluate the given quantities assuming that $$ \begin{array}{l} \log _{3} x=5.3 \text { and } \log _{3} y=2.1 \\ \log _{4} u=3.2 \text { and } \log _{4} v=1.3 \end{array} $$ $$ \log _{4} \frac{u^{2}}{v^{3}} $$

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