Question

which of the following best explains how to find the value of $8^{\frac{5}{3}}$?
answer
in the table, each time $8^x$ doubles, the exponent is multiplied by $1/3$
a number with the exponent of $5/3$ is the same as $5$ times the cube root of the number.
in the table, each $x$ increases by $1/3$, the value of $8^x$ is multiplied by $8$
a number with an exponent of $5/3$ is the same as the cube root of the number raised to the $5$th power
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Explanation:

To find \(8^{\frac{5}{3}}\), we use the exponent rule \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\) (or equivalently \(a^{\frac{m}{n}} = (\sqrt[n]{a})^m\)). For \(8^{\frac{5}{3}}\), this means taking the cube root of \(8^5\) (or taking the cube root of 8 first and then raising it to the 5th power). Let's analyze the options:

• The first option about doubling \(8^x\) and exponent multiplied by \(1/3\) is incorrect as it misinterprets the relationship.
• The second option says "5 times the cube root of the number", but \(8^{\frac{5}{3}}=(\sqrt[3]{8})^5 = 2^5=32\) or \(\sqrt[3]{8^5}=\sqrt[3]{32768} = 32\), while "5 times the cube root of 8" is \(5\times2 = 10\), which is wrong.
• The third option about \(x\) increasing by \(1/3\) and \(8^x\) multiplied by 8 is incorrect as the base - exponent relationship here is not about that pattern.
• The fourth option: A number with an exponent of \(\frac{5}{3}\) is the same as the cube root of the number raised to the 5th power. Using the exponent rule \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\), for \(a = 8\), \(m = 5\), \(n=3\), we have \(8^{\frac{5}{3}}=\sqrt[3]{8^5}\), which is the cube root of 8 raised to the 5th power.

Answer:

A number with an exponent of \( \frac{5}{3} \) is the same as the cube root of the number raised to the 5th power