Chapter 7.5, Problem 31P

Simplify each expression using the properties of exponents, and then write the expression in radical form.

$\left(2x^3\right)\left(4x^{\frac{1}{3}}\right)$

To determine

To write simply the expression $\left(2x^3\right)\left(4x^{\frac{1}{3}}\right)$ in radical form.

Answer to Problem 31P

  $\left(2x^3\right)\left(4x^{\frac{1}{3}}\right)=8\sqrt[3]{x^{10}}$

Explanation of Solution

Given information :

Given expression $\left(2x^3\right)\left(4x^{\frac{1}{3}}\right)$

  $\begin{array}{l}\text{Since}{a}^{\frac{1}{n}}=\sqrt[n]{a}\mathrm{............}(i)and{\left(a^m\right)}^n=a^{m\times n}\mathrm{........}\text{(ii)}and{\left(ab\right)}^m=a^m\times b^m\mathrm{.......}(iii)\\ a^m\times a^n=a^{m+n}\mathrm{.........}(iv)\end{array}$

Now consider $\left(2x^3\right)\left(4x^{\frac{1}{3}}\right)=2\left(4\right)\left(x^3\times x^{\frac{1}{3}}\right)$

  $\begin{array}{l}=8x^{3+\frac{1}{3}}\text{from}\text{rule}(iv)\\ =8x^{\frac{9+1}{3}}\\ =8x^{\frac{10}{3}}\\ =8\sqrt[3]{x^{10}}\text{from}\text{rule}(i)\\ \therefore \left(2x^3\right)\left(4x^{\frac{1}{3}}\right)=8\sqrt[3]{x^{10}}\end{array}$

Hence, $\left(2x^3\right)\left(4x^{\frac{1}{3}}\right)=8\sqrt[3]{x^{10}}$