Question

Evaluate each exponential. $$ \left(\frac{16}{81}\right)^{-3 / 4} $$

   Evaluate each exponential.
$$
\left(\frac{16}{81}\right)^{-3 / 4}
$$

Intermediate Algebra

Intermediate Algebra

John Hornsby; Terry… 13th Edition

Chapter 7, Problem 39

Instant Answer

Solved by Expert Rahul Mittal

02/15/2022

Step 1

So, we can rewrite the given expression as $\left(\frac{81}{16}\right)^{3 / 4}$. Show more…

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Evaluate each exponential. $$ \left(\frac{16}{81}\right)^{-3 / 4} $$

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Key Concepts

Rational Exponents

Rational exponents express both roots and powers. An exponent in the form m/n means the nth root of a number raised to the m power, or equivalently, taking the power first and then the nth root. This interpretation bridges the gap between exponential and radical notation.

Negative Exponents

Negative exponents indicate reciprocals. When a number with a negative exponent is encountered, it can be rewritten by taking the reciprocal of the base and changing the exponent to positive. This property simplifies the evaluation of expressions by converting them into forms that are easier to manipulate.

Exponentiation of Fractions

When an expression that is a fraction is raised to an exponent, the exponent is applied to both the numerator and the denominator. This rule enables the systematic evaluation of expressions where both the numerator and denominator are influenced by the exponent, maintaining the relationship between them.

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