/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 105 Simplify the expressions. $$ ... [FREE SOLUTION] | 91影视

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Chapter 0: Problem 105

Simplify the expressions. $$ \left(\frac{x}{y}\right)^{1 / 3}\left(\frac{y}{x}\right)^{2 / 3} $$

Short Answer

Expert verified
The simplified expression is \(\frac{y^{\frac{1}{3}}}{x^{\frac{1}{3}}}\).

Step by step solution

01

Combine the exponents

Since both expressions have the same bases (\(\frac{x}{y}\) and \(\frac{y}{x}\), respectively), we can multiply them as follows: \[ \left(\frac{x}{y}\right)^{1/3} \cdot \left(\frac{y}{x}\right)^{2/3}. \]
02

Apply the exponent rule 饾憥^(饾憵) * 饾憦^(饾憵) = (饾憥 * 饾憦)^(饾憵)

Rewrite the expression as: \[ \left(\frac{x^{\frac{1}{3}}}{y^{\frac{1}{3}}} \cdot \frac{y^{\frac{2}{3}}}{x^{\frac{2}{3}}}\right). \]
03

Simplify the expression

Combine the terms: \[ \frac{x^{\frac{1}{3}}}{y^{\frac{1}{3}}} \cdot \frac{y^{\frac{2}{3}}}{x^{\frac{2}{3}}} = \frac{x^{\frac{1}{3}} y^{\frac{2}{3}}}{y^{\frac{1}{3}} x^{\frac{2}{3}}}, \]
04

Apply the exponent rule 饾憥^(饾憵) * 饾憥^(饾憶) = 饾憥^(饾憵+饾憶)

Cancel the common terms to simplify the expression further: \[ = \frac{y^{\frac{2}{3} - \frac{1}{3}}}{x^{\frac{2}{3} - \frac{1}{3}}} = \frac{y^{\frac{1}{3}}}{x^{\frac{1}{3}}}, \]
05

Write the final simplified expression

The final simplified expression is: \[ \frac{y^{\frac{1}{3}}}{x^{\frac{1}{3}}}. \] In conclusion, the simplified expression of the given expression is \(\frac{y^{\frac{1}{3}}}{x^{\frac{1}{3}}}\).

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Most popular questions from this chapter

Convert each expression in Exercises into its technology formula equivalent as in the table in the text.\(\frac{2}{3+x}-x y^{2}\)

Convert each expression in Exercises into its technology formula equivalent as in the table in the text.\(2^{2 x^{2}-x}+1\)

Calculate each expression in Exencises giving the answer as \(a\) whole mumber or a fraction in lowest terms.\(3\left(\frac{1}{9}-\left(\frac{2}{3}\right)^{2}\right)^{2}+1\)

Convert each expression in Exercises into its technology formula equivalent as in the table in the text.\(3^{2 x-1}+4^{x}-1\)

By any method, determine all possible real solutions of each equation Check your answers by substitution. \(x^{4}+x^{2}-1=0\)
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