example 5
evaluate each expression.
19. $27^{\frac{1}{3}}$
20. $256^{\frac{1}{4}}$
21. $16^{-\frac{1}{2}}$
22. $81^{-\frac{1}{4}}$
23. $1024^{\frac{3}{5}}$
24. $16^{-\frac{3}{4}}$

Question

Accepted Answer

### Problem 19: $ 27^{\frac{1}{3}} $
# Explanation:
## Step 1: Recall the definition of rational exponents
A rational exponent $ a^{\frac{1}{n}} $ means the $ n $-th root of $ a $, i.e., $ a^{\frac{1}{n}}=\sqrt[n]{a} $. So for $ 27^{\frac{1}{3}} $, we need to find the cube root of 27.
## Step 2: Calculate the cube root of 27
We know that $ 3\times3\times3 = 3^{3}=27 $, so $ \sqrt[3]{27}=3 $.
# Answer:
$ 3 $

### Problem 20: $ 256^{\frac{1}{4}} $
# Explanation:
## Step 1: Recall the definition of rational exponents
Using the rule $ a^{\frac{1}{n}}=\sqrt[n]{a} $, for $ 256^{\frac{1}{4}} $, we need to find the fourth root of 256.
## Step 2: Calculate the fourth root of 256
We know that $ 4\times4\times4\times4=4^{4} = 256 $, so $ \sqrt[4]{256}=4 $.
# Answer:
$ 4 $

### Problem 21: $ 16^{-\frac{1}{2}} $
# Explanation:
## Step 1: Recall the negative exponent rule
The negative exponent rule states that $ a^{-n}=\frac{1}{a^{n}} $, so $ 16^{-\frac{1}{2}}=\frac{1}{16^{\frac{1}{2}}} $.
## Step 2: Recall the definition of rational exponents
Using $ a^{\frac{1}{n}}=\sqrt[n]{a} $, $ 16^{\frac{1}{2}}=\sqrt{16} $.
## Step 3: Calculate the square root of 16
We know that $ 4\times4 = 16 $, so $ \sqrt{16}=4 $. Then $ \frac{1}{16^{\frac{1}{2}}}=\frac{1}{4} $.
# Answer:
$ \frac{1}{4} $

### Problem 22: $ 81^{-\frac{1}{4}} $
# Explanation:
## Step 1: Recall the negative exponent rule
Using $ a^{-n}=\frac{1}{a^{n}} $, we have $ 81^{-\frac{1}{4}}=\frac{1}{81^{\frac{1}{4}}} $.
## Step 2: Recall the definition of rational exponents
Using $ a^{\frac{1}{n}}=\sqrt[n]{a} $, $ 81^{\frac{1}{4}}=\sqrt[4]{81} $.
## Step 3: Calculate the fourth root of 81
We know that $ 3\times3\times3\times3 = 3^{4}=81 $, so $ \sqrt[4]{81}=3 $. Then $ \frac{1}{81^{\frac{1}{4}}}=\frac{1}{3} $.
# Answer:
$ \frac{1}{3} $

### Problem 23: $ 1024^{\frac{3}{5}} $
# Explanation:
## Step 1: Recall the rule for rational exponents $ a^{\frac{m}{n}}=\sqrt[n]{a^{m}}=(\sqrt[n]{a})^{m} $
So we can rewrite $ 1024^{\frac{3}{5}} $ as $ (\sqrt[5]{1024})^{3} $ or $ \sqrt[5]{1024^{3}} $. We'll use $ (\sqrt[5]{1024})^{3} $ for simplicity.
## Step 2: Find the fifth root of 1024
We know that $ 4\times4\times4\times4\times4=4^{5} = 1024 $, so $ \sqrt[5]{1024}=4 $.
## Step 3: Raise the result to the power of 3
$ 4^{3}=4\times4\times4 = 64 $.
# Answer:
$ 64 $

### Problem 24: $ 16^{-\frac{3}{4}} $
# Explanation:
## Step 1: Recall the negative exponent rule
Using $ a^{-n}=\frac{1}{a^{n}} $, we have $ 16^{-\frac{3}{4}}=\frac{1}{16^{\frac{3}{4}}} $.
## Step 2: Recall the rule for rational exponents $ a^{\frac{m}{n}}=(\sqrt[n]{a})^{m} $
So $ 16^{\frac{3}{4}}=(\sqrt[4]{16})^{3} $.
## Step 3: Find the fourth root of 16
We know that $ 2\times2\times2\times2=2^{4} = 16 $, so $ \sqrt[4]{16}=2 $.
## Step 4: Raise the result to the power of 3
$ 2^{3}=8 $. Then $ \frac{1}{16^{\frac{3}{4}}}=\frac{1}{8} $.
# Answer:
$ \frac{1}{8} $

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QUESTION IMAGE

Question

Explanation:

Problem 19: \( 27^{\frac{1}{3}} \)

Step 1: Recall the definition of rational exponents

Step 2: Calculate the cube root of 27

Step 1: Recall the definition of rational exponents

Step 2: Calculate the fourth root of 256

Step 1: Recall the negative exponent rule

Step 2: Recall the definition of rational exponents

Step 3: Calculate the square root of 16

Answer:

Problem 20: \( 256^{\frac{1}{4}} \)