Chapter 3, Problem 2VE

Solution

Two different pairs of like radicals.

  $\begin{array}{l}2\sqrt{3}\text{ and 5}\sqrt{3},\\ \sqrt[3]{4}\text{}\text{ and 3}\sqrt[3]{4}\end{array}$

Given:

The pairs of like radicals.

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: $\frac{a^m}{a^n}=a^{m-n}$
• Product property of radicals: $\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}$
• Quotient property of radicals: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
• Sum of like radicals’ property: $p\sqrt[n]{a}+q\sqrt[n]{a}=\left(p+q\right)\sqrt[n]{a}$
• The exponent form of the nth root of a number a is given by $\sqrt[n]{a}=a^{\frac{1}{n}}$

Calculation:

Recall that the index of the radical $\sqrt[n]{a}$ is n . The index of a radical is the number appearing on the front top of the radical symbol.

The like radicals have the same index as well as the same radicand inside the radicals.

Thus, two pairs of the like radicals are $2\sqrt{3}\text{, 5}\sqrt{3},\text{ and }\sqrt[3]{4}\text{},\text{ 3}\sqrt[3]{4}$ .