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Chapter 7: Problem 51

Write each exponential as a radical. Assume that all variables represent positive real numbers. Use the definition that takes the root first. $$ (2 y+x)^{2 / 3} $$

Short Answer

Expert verified
\( (2y + x)^{2/3} = \sqrt[3]{(2y + x)^2} \)

Step by step solution

01

- Identify the exponent

The given expression is \((2y + x)^{2/3}\). Here, the exponent is \[ \frac{2}{3} \].
02

- Express the exponent in radical form

Recall the definition: for a rational exponent \[ \frac{m}{n} \], the base \[ a \] raised to the power can be written as \((a^{m/n} = \sqrt[n]{a^{m}})\).
03

- Apply the definition

Using the definition, \((2y + x)^{2/3}\) can be written as \[ \sqrt[3]{(2y + x)^{2}} \].
04

- Write the final answer

Hence, the exponential \((2 y+x)^{2 / 3}\) can be written as \sqrt[3]{(2 y+x)^{2}}.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Exponents
Rational exponents combine the concepts of exponents and roots in a single notation. When we see an expression like \(a^{m/n}\), it means we are dealing with both a power and a root.
Here's how to understand it:
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Most popular questions from this chapter

Write each radical as an exponential and simplify. Leave answers in exponential form. Assume that all variables represent positive numbers. $$ \sqrt[5]{x^{3}} \cdot \sqrt[4]{x} $$

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{12}{\sqrt{6}+\sqrt{3}} $$

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{\sqrt{8}}{3-\sqrt{2}} $$

Simplify. Assume that all variables represent positive real numbers. \(\sqrt{169 s^{5} t^{10}}\)

Write each quotient in lowest terms. Assume that all variables represent positive real numbers. $$ \frac{3-3 \sqrt{5}}{3} $$
See all solutions

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