https://file.aiquickdraw.com/mathbot/file/1729033050607uebrawk5.jpg

Let's break down and solve the expression provided:

We are asked to evaluate \( 32^{\frac{4}{5}} \times 36^{-\frac{1}{2}} \).

### Step 1: Simplify each term
#### 1. \( 32^{\frac{4}{5}} \)
The base \( 32 \) can be written as \( 32 = 2^5 \). So we have:
\[
32^{\frac{4}{5}} = (2^5)^{\frac{4}{5}} = 2^{5 \times \frac{4}{5}} = 2^4 = 16
\]

#### 2. \( 36^{-\frac{1}{2}} \)
The exponent \( -\frac{1}{2} \) indicates a reciprocal and a square root:
\[
36^{-\frac{1}{2}} = \frac{1}{36^{\frac{1}{2}}} = \frac{1}{\sqrt{36}} = \frac{1}{6}
\]

### Step 2: Multiply the results
Now we multiply the two simplified terms:
\[
32^{\frac{4}{5}} \times 36^{-\frac{1}{2}} = 16 \times \frac{1}{6} = \frac{16}{6}
\]

### Step 3: Simplify the fraction
\[
\frac{16}{6} = \frac{8}{3}
\]

Thus, the final answer is \( \frac{8}{3} \).

Do you need any more details or have any questions?

---

Here are 5 related questions you can explore:
1. How do you convert between fractional and negative exponents?
2. What is the general rule for simplifying roots of powers?
3. What happens when the base of a number has a negative exponent?
4. Can you solve \( 27^{\frac{2}{3}} \times 81^{-\frac{1}{4}} \)?
5. How would you calculate \( 49^{\frac{3}{2}} \)?

**Tip:** When dealing with fractional exponents, remember that the numerator represents a power, and the denominator represents a root!

Work out 32^(4/5) * 36^(-1/2). Give your answer as an integer or as a fraction in its simplest form.

This problem explores how to simplify 32^(4/5) * 36^(-1/2) using the laws of exponents, fractional exponents, and negative exponents. The solution includes key concepts such as simplifying roots and powers, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation