To compute \( 81^{-\frac{3}{4}} \), let's break it down step by step:

1. First, recall that a fractional exponent can be interpreted as a combination of a root and a power:
   \[
   81^{-\frac{3}{4}} = \left(81^{\frac{1}{4}}\right)^{-3}
   \]
   So, we need to find the fourth root of 81.

2. The fourth root of 81 is:
   \[
   81^{\frac{1}{4}} = \sqrt[4]{81} = \sqrt[4]{3^4} = 3
   \]
   
3. Now raise the result to the power of -3:
   \[
   3^{-3} = \frac{1}{3^3} = \frac{1}{27}
   \]

So, \( 81^{-\frac{3}{4}} = \frac{1}{27} \).

Would you like more details or have any questions?

Here are 5 related questions to expand on this:
1. What is \( 81^{\frac{1}{4}} \) if calculated directly?
2. How do you calculate the fourth root of a number like 16?
3. What happens when you raise a number to a negative power?
4. How can you simplify expressions with fractional exponents?
5. How do roots and powers relate in algebra?

**Tip:** Negative exponents represent reciprocal values. For example, \( a^{-n} = \frac{1}{a^n} \).

Learn how to compute 81 raised to the power of -3/4. This detailed explanation covers fractional exponents, roots, and negative powers. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation