Math / Algebra

QuestionLesson 4 Practice Problems
1. Evaluate $8^{\frac{5}{3}}$ .
2. Select all expressions that are equal to $64^{\frac{3}{2}}$ . A. 96 B. $8^3$ C. 512 D. $4^2$ E. $\sqrt{64^3}$ F. $\sqrt[3]{64}^2$

Studdy Solution

STEP 1

What is this asking? We need to calculate $8$ raised to the power of five-thirds, and figure out which expressions are the same as $64$ raised to the power of three-halves. Watch out! Remember fractional exponents have two parts: the numerator tells us the power, and the denominator tells us the root.
Don't mix them up!

STEP 2

1. Evaluate the first expression.
2. Evaluate each option for the second question.

STEP 3

We're asked to evaluate $8^{\frac{5}{3}}$ .
Remember, a fractional exponent like $\frac{5}{3}$ means we take the 3rd root and raise it to the 5th power.
We can rewrite this as $\left(\sqrt[3]{8}\right)^5$ .
This makes it easier to calculate!

STEP 4

The cube root of 8 is 2, because $2 \cdot 2 \cdot 2 = 8$ .
So, $\sqrt[3]{8} = 2$ .
Now we have $2^5$ .

STEP 5

Now, we just need to calculate $2^5$ , which means multiplying 2 by itself five times: $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$ .
So, $8^{\frac{5}{3}} = 32$ .

STEP 6

Option A is just the number 96.
Let's see what $64^{\frac{3}{2}}$ actually is.
We can rewrite $64^{\frac{3}{2}}$ as $(\sqrt{64})^3$ .
The square root of 64 is 8, so we have $8^3$ . $8 \cdot 8 \cdot 8 = 512$ .
So, option A is incorrect because $96 \ne 512$ .

STEP 7

Option B is $8^3$ .
We just figured out that $64^{\frac{3}{2}} = 8^3$ , which is 512.
So, option B is correct!

STEP 8

Option C is 512.
We calculated that $64^{\frac{3}{2}} = 512$ , so option C is correct!

STEP 9

Option D is $4^2$ .
This is $4 \cdot 4 = 16$ .
Since $16 \ne 512$ , option D is incorrect.

STEP 10

Option E is $\sqrt{64^3}$ .
This is the same as $(64^3)^{\frac{1}{2}}$ , which is the same as $64^{\frac{3}{2}}$ using the power of a power rule!
So, option E is correct!

STEP 11

Option F is $\sqrt[3]{64}^2$ .
The cube root of 64 is 4 (because $4 \cdot 4 \cdot 4 = 64$ ).
So, we have $4^2$ , which is $4 \cdot 4 = 16$ .
Since $16 \ne 512$ , option F is incorrect.

STEP 12

$8^{\frac{5}{3}}$ is equal to 32.
The expressions equal to $64^{\frac{3}{2}}$ are B ( $8^3$ ), C (512), and E ( $\sqrt{64^3}$ ).

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QuestionLesson 4 Practice Problems
1. Evaluate $8^{\frac{5}{3}}$ .
2. Select all expressions that are equal to $64^{\frac{3}{2}}$ . A. 96 B. $8^3$ C. 512 D. $4^2$ E. $\sqrt{64^3}$ F. $\sqrt[3]{64}^2$

Studdy Solution

STEP 1

STEP 2

1. Evaluate the first expression.
2. Evaluate each option for the second question.

STEP 3

We're asked to evaluate $8^{\frac{5}{3}}$ .
Remember, a fractional exponent like $\frac{5}{3}$ means we take the 3rd root and raise it to the 5th power.
We can rewrite this as $\left(\sqrt[3]{8}\right)^5$ .
This makes it easier to calculate!

STEP 4

The cube root of 8 is 2, because $2 \cdot 2 \cdot 2 = 8$ .
So, $\sqrt[3]{8} = 2$ .
Now we have $2^5$ .

STEP 5

Now, we just need to calculate $2^5$ , which means multiplying 2 by itself five times: $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$ .
So, $8^{\frac{5}{3}} = 32$ .

STEP 6

Option A is just the number 96.
Let's see what $64^{\frac{3}{2}}$ actually is.
We can rewrite $64^{\frac{3}{2}}$ as $(\sqrt{64})^3$ .
The square root of 64 is 8, so we have $8^3$ . $8 \cdot 8 \cdot 8 = 512$ .
So, option A is incorrect because $96 \ne 512$ .

STEP 7

Option B is $8^3$ .
We just figured out that $64^{\frac{3}{2}} = 8^3$ , which is 512.
So, option B is correct!

STEP 8

Option C is 512.
We calculated that $64^{\frac{3}{2}} = 512$ , so option C is correct!

STEP 9

Option D is $4^2$ .
This is $4 \cdot 4 = 16$ .
Since $16 \ne 512$ , option D is incorrect.

STEP 10

Option E is $\sqrt{64^3}$ .
This is the same as $(64^3)^{\frac{1}{2}}$ , which is the same as $64^{\frac{3}{2}}$ using the power of a power rule!
So, option E is correct!

STEP 11

Option F is $\sqrt[3]{64}^2$ .
The cube root of 64 is 4 (because $4 \cdot 4 \cdot 4 = 64$ ).
So, we have $4^2$ , which is $4 \cdot 4 = 16$ .
Since $16 \ne 512$ , option F is incorrect.

STEP 12

$8^{\frac{5}{3}}$ is equal to 32.
The expressions equal to $64^{\frac{3}{2}}$ are B ( $8^3$ ), C (512), and E ( $\sqrt{64^3}$ ).

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QuestionLesson 4 Practice Problems1. Evaluate 8538^{\frac{5}{3}}835​.2. Select all expressions that are equal to 643264^{\frac{3}{2}}6423​. A. 96 B. 838^383 C. 512 D. 424^242 E. 643\sqrt{64^3}643​ F. 6432\sqrt[3]{64}^2364​2

Studdy Solution

STEP 1

STEP 2

1. Evaluate the first expression.2. Evaluate each option for the second question.

STEP 3

We're asked to evaluate 8538^{\frac{5}{3}}835​.Remember, a fractional exponent like 53\frac{5}{3}35​ means we take the **3rd root** and raise it to the **5th power**.We can rewrite this as (83)5\left(\sqrt[3]{8}\right)^5(38​)5.This makes it easier to calculate!

STEP 4

The cube root of **8** is **2**, because 2⋅2⋅2=82 \cdot 2 \cdot 2 = 82⋅2⋅2=8.So, 83=2\sqrt[3]{8} = 238​=2.Now we have 252^525.

STEP 5

Now, we just need to calculate 252^525, which means multiplying **2** by itself **five** times: 2⋅2⋅2⋅2⋅2=322 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 322⋅2⋅2⋅2⋅2=32.So, 853=328^{\frac{5}{3}} = 32835​=32.

STEP 6

STEP 7

Option B is 838^383.We just figured out that 6432=8364^{\frac{3}{2}} = 8^36423​=83, which is **512**.So, option B is **correct**!

STEP 8

Option C is **512**.We calculated that 6432=51264^{\frac{3}{2}} = 5126423​=512, so option C is **correct**!

STEP 9

Option D is 424^242.This is 4⋅4=164 \cdot 4 = 164⋅4=16.Since 16≠51216 \ne 51216=512, option D is **incorrect**.

STEP 10

Option E is 643\sqrt{64^3}643​.This is the same as (643)12(64^3)^{\frac{1}{2}}(643)21​, which is the same as 643264^{\frac{3}{2}}6423​ using the power of a power rule!So, option E is **correct**!

STEP 11

Option F is 6432\sqrt[3]{64}^2364​2.The cube root of **64** is **4** (because 4⋅4⋅4=644 \cdot 4 \cdot 4 = 644⋅4⋅4=64).So, we have 424^242, which is 4⋅4=164 \cdot 4 = 164⋅4=16.Since 16≠51216 \ne 51216=512, option F is **incorrect**.

STEP 12

8538^{\frac{5}{3}}835​ is equal to **32**.The expressions equal to 643264^{\frac{3}{2}}6423​ are B (838^383), C (**512**), and E (643\sqrt{64^3}643​).

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QuestionLesson 4 Practice Problems
1. Evaluate $8^{\frac{5}{3}}$ .
2. Select all expressions that are equal to $64^{\frac{3}{2}}$ . A. 96 B. $8^3$ C. 512 D. $4^2$ E. $\sqrt{64^3}$ F. $\sqrt[3]{64}^2$

1. Evaluate the first expression.
2. Evaluate each option for the second question.

We're asked to evaluate $8^{\frac{5}{3}}$ .
Remember, a fractional exponent like $\frac{5}{3}$ means we take the 3rd root and raise it to the 5th power.
We can rewrite this as $\left(\sqrt[3]{8}\right)^5$ .
This makes it easier to calculate!

The cube root of 8 is 2, because $2 \cdot 2 \cdot 2 = 8$ .
So, $\sqrt[3]{8} = 2$ .
Now we have $2^5$ .

Now, we just need to calculate $2^5$ , which means multiplying 2 by itself five times: $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$ .
So, $8^{\frac{5}{3}} = 32$ .

Option B is $8^3$ .
We just figured out that $64^{\frac{3}{2}} = 8^3$ , which is 512.
So, option B is correct!

Option C is 512.
We calculated that $64^{\frac{3}{2}} = 512$ , so option C is correct!

Option D is $4^2$ .
This is $4 \cdot 4 = 16$ .
Since $16 \ne 512$ , option D is incorrect.

Option E is $\sqrt{64^3}$ .
This is the same as $(64^3)^{\frac{1}{2}}$ , which is the same as $64^{\frac{3}{2}}$ using the power of a power rule!
So, option E is correct!

Option F is $\sqrt[3]{64}^2$ .
The cube root of 64 is 4 (because $4 \cdot 4 \cdot 4 = 64$ ).
So, we have $4^2$ , which is $4 \cdot 4 = 16$ .
Since $16 \ne 512$ , option F is incorrect.

$8^{\frac{5}{3}}$ is equal to 32.
The expressions equal to $64^{\frac{3}{2}}$ are B ( $8^3$ ), C (512), and E ( $\sqrt{64^3}$ ).