Question

Simplify each expression. Express the answer so that all exponents are positive. Whenever an exponent is 0 or negative, we assume that the base is not \(0 .\) \(\left(\frac{5 x^{-2}}{6 y^{-2}}\right)^{-3}\)

Short answer

\(\frac{216 x^6}{125 y^6}\)

Step-by-step solution

Distribute the negative exponent

Use the rule \( \left( \frac{a}{b} \right)^{n} = \frac{a^n}{b^n} \) to distribute the -3 exponent to both the numerator and the denominator: \[ \left( \frac{5 x^{-2}}{6 y^{-2}} \right)^{-3} = \frac{\left( 5 x^{-2} \right)^{-3}}{\left( 6 y^{-2} \right)^{-3}} \]

Simplify the numerator and denominator

Apply the power rule \( (a^m)^n = a^{m \cdot n} \) to both the numerator and the denominator terms: \[ \frac{\left( 5^{-3} \cdot (x^{-2})^{-3} \right)}{\left( 6^{-3} \cdot (y^{-2})^{-3} \right)} = \frac{5^{-3} \cdot x^{6}}{6^{-3} \cdot y^{6}} \]

Simplify the constants

Calculate the constants with negative exponents: \[ 5^{-3} = \frac{1}{5^3} = \frac{1}{125} \quad \text{and} \quad 6^{-3} = \frac{1}{6^3} = \frac{1}{216} \] Rewrite the expression: \[ \frac{\frac{1}{125} \cdot x^6}{\frac{1}{216} \cdot y^6} = \frac{216 \cdot x^6}{125 \cdot y^6} \]

Combine the expression

Put everything together to write the expression in its simplest form: \[ \left( \frac{5 x^{-2}}{6 y^{-2}} \right)^{-3} = \frac{216 x^6}{125 y^6} \]

Key concepts

Negative Exponents

When you see an expression with a negative exponent, remember it signifies division and not multiplication. For example, if you have an exponent like \( a^{-n} \), it can be rewritten as \( \frac{1}{a^n} \). Applying this rule, we rewrite terms with negative exponents in the original exercise: \(x^{-2} \) becomes \( \frac{1}{x^2} \) and \( y^{-2} \) becomes \( \frac{1}{y^2} \). This makes it easier to simplify complex expressions that contain negative exponents. Importantly, a fraction raised to a negative power can be converted using \( \frac{a}{b}^{-n} = \frac{b^n}{a^n} \). Always convert negative exponents to positive as your final step.

Power Rule

The power rule, or more formally the exponentiation rule, is an essential exponent rule that simplifies expressions where an exponent is raised to another exponent. Specifically, \( (a^m)^n = a^{m \cdot n} \). You see its application in Step 2 of the original solution: \( (x^{-2})^{-3} = x^{6} \) and \( (y^{-2})^{-3} = y^{6} \). By applying the power rule, you multiply the exponents together for both the numerator and the denominator. This simplifies the expression significantly, as demonstrated in the original problem solution.

Division of Exponents

Division of exponents can be initially confusing, but a simple rule guides you: when you divide like bases with exponents, you subtract the exponents. So, \( \frac{a^m}{a^n} = a^{m-n} \). This isn’t directly used in our problem but is an important concept to grasp so you know how to handle division of terms with equal bases. Conversion of terms with negative exponents also often involves division, exemplified in our original problem when positive exponents were isolated through division transformations. Combining constants with their respective negative exponents also uses foundations of division: \( 5^{-3} \) converts to \( \frac{1}{5^3} \) and equally for 6.