Chapter 3.1, Problem 46E

Solution

The expression that has the greatest value.

  $\left(B\right)5^{\frac{3}{2}}$

Given:

The expressions

  $\begin{array}{l}\left(A\right){27}^{\frac{3}{5}}\\ \left(B\right)5^{\frac{3}{2}}\\ \left(C\right)\sqrt[3]{81}\\ \left(D\right){\left(\sqrt[3]{2}\right)}^8\end{array}$

Concept Used:

• The even inter root of a number is defined for positive values only that are greater than or equal to 0.
• The odd integer root of all real numbers is defined.
• Power of a power property: ${\left(x^a\right)}^b=x^{a\cdot b}$
• Exponent property, $a^{-n}=\frac{1}{a^n}$
• The exponent form of the nth root of a number a is given by $\sqrt[n]{a}=a^{\frac{1}{n}}$

Calculation:

In order to determine the expression that has the greatest vale, use the calculator evaluate each expression as shown below

  $\begin{array}{l}A.{27}^{\frac{3}{5}}\approx 7.22\\ B.5^{\frac{3}{2}}\approx 11.18\\ C.\sqrt[3]{81}\approx 4.33\\ D.{\left(\sqrt[3]{2}\right)}^8\approx 6.35\end{array}$

Therefore, the expression $5^{\frac{3}{2}}$ has the greatest value.