Chapter 3.2, Problem 1E

Solution

Whether $2\sqrt{5}\text{ and }2\sqrt[3]{5}$ are like radicals or not.

No

Given:

The radical expressions $2\sqrt{5}\text{ and }2\sqrt[3]{5}$ .

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: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
• 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.

Since the index of the radical term $2\sqrt{5}$ is 2 while the index of the radical term $2\sqrt[3]{5}$ is 3, the given radical terms are not like radicals.

Therefore, the expressions $2\sqrt{5}\text{ and }2\sqrt[3]{5}$ are not like radicals.