Chapter 3.2, Problem 1GP

Solution

To simplify the expression ${\left(5^{\frac{1}{3}}\cdot 7^{\frac{1}{4}}\right)}^3$ .

  $5\cdot 7^{\frac{3}{4}}$

Given:

The expression ${\left(5^{\frac{1}{3}}\cdot 7^{\frac{1}{4}}\right)}^3$ .

Concept Used:

• Power of a power property: ${\left(x^a\right)}^b=x^{a\cdot b}$
• Product of power property: $a^m\cdot a^n=a^{m+n}$
• Quotient of power property: $\frac{a^m}{a^n}=a^{m-n}$
• 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 simplify the given expression ${\left(5^{\frac{1}{3}}\cdot 7^{\frac{1}{4}}\right)}^3$ , use the exponent properties and simplify further as shown below,

  $\begin{array}{c}{\left(5^{\frac{1}{3}}\cdot 7^{\frac{1}{4}}\right)}^3=5^{\frac{1}{3}\cdot 3}\cdot 7^{\frac{1}{4}\cdot 3}\\ =5^{\frac{3}{3}}\cdot 7^{\frac{3}{4}}\\ =5^1\cdot 7^{\frac{3}{4}}\\ =5\cdot 7^{\frac{3}{4}}\end{array}$

Therefore, the simplified expression is $5\cdot 7^{\frac{3}{4}}$ .