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Chapter 0: Problem 106

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

Short Answer

Expert verified
The simplified expression is \(1\).

Step by step solution

01

Write down the expression

We begin by writing down the expression given: \( \left(\frac{x}{y}\right)^{-1/3}\left(\frac{y}{x}\right)^{1/3} \)
02

Use exponent rules to rewrite the expression

Recall that \(\left( AB \right)^{n} = A^n B^n\). Using this rule, we can rewrite the expression: \( \left(\frac{x}{y}\right)^{-1/3}\left(\frac{y}{x}\right)^{1/3} = \left(\frac{x^{-1/3}}{y^{-1/3}}\right)\left(\frac{y^{1/3}}{x^{1/3}}\right) \)
03

Combine the fractions

Next, we multiply the two fractions by multiplying the numerators together and the denominators together: \( \left(\frac{x^{-1/3}}{y^{-1/3}}\right)\left(\frac{y^{1/3}}{x^{1/3}}\right) = \frac{x^{-1/3}y^{1/3}}{y^{-1/3}x^{1/3}} \)
04

Simplify the exponents

Recall that if \(a\) and \(b\) have the same base, then \(a^m \cdot a^n = a^{m + n}\). We can use this rule to simplify the exponents of our expression: \( \frac{x^{-1/3}y^{1/3}}{y^{-1/3}x^{1/3}} = \frac{x^{-1/3 + 1/3}y^{1/3 + -1/3}}{1} = \frac{x^0y^0}{1} \)
05

Simplify the expression

Since \(x^0 = y^0 = 1\), the expression simplifies to 1: \( \frac{x^0y^0}{1} = 1 \) The simplified expression is \(1\).

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