Solved on Nov 08, 2023

Simplify the expression $\left(16 b^{\frac{1}{3}}\right)^{\frac{3}{2}}$ as a power of $b$ .

STEP 1

Assumptions1. We are given the expression $\left(16 b^{\frac{1}{3}}\right)^{\frac{3}{}}$ . . We want to write this expression as a power.

STEP 2

First, we need to understand that when we raise a power to another power, we multiply the exponents. This is a basic rule of exponents. $\left(a^m\right)^n = a^{m \times n}$

STEP 3

Apply the rule of exponents to the given expression. We have two parts in the expression,16 and $b^{\frac{1}{3}}$ , and both are raised to the power of $\frac{3}{2}$ .
$\left(16 b^{\frac{1}{3}}\right)^{\frac{3}{2}} =16^{\frac{3}{2}} \times \left(b^{\frac{1}{3}}\right)^{\frac{3}{2}}$

STEP 4

Now, calculate the power of16. $16^{\frac{3}{2}} = \sqrt{16^3}$

STEP 5

Calculate the value of $16^3$ .
$16^3 =409$

STEP 6

Now, calculate the square root of4096.
$\sqrt{4096} =64$

STEP 7

Now, calculate the power of $b^{\frac{1}{3}}$ .
$\left(b^{\frac{1}{3}}\right)^{\frac{3}{2}} = b^{\frac{1}{3} \times \frac{3}{2}}$

STEP 8

Multiply the exponents.
$b^{\frac{1}{3} \times \frac{3}{2}} = b^{\frac{1}{2}}$

STEP 9

Now, combine the results from steps6 and8 to write the original expression as a power.
$\left(16 b^{\frac{}{3}}\right)^{\frac{3}{2}} =64b^{\frac{}{2}}$ So, $\left(16 b^{\frac{}{3}}\right)^{\frac{3}{2}}$ as a power is $64b^{\frac{}{2}}$ .

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Simplify the expression $\left(16 b^{\frac{1}{3}}\right)^{\frac{3}{2}}$ as a power of $b$ .

STEP 1

Assumptions1. We are given the expression $\left(16 b^{\frac{1}{3}}\right)^{\frac{3}{}}$ . . We want to write this expression as a power.

STEP 2

First, we need to understand that when we raise a power to another power, we multiply the exponents. This is a basic rule of exponents. $\left(a^m\right)^n = a^{m \times n}$

STEP 3

Apply the rule of exponents to the given expression. We have two parts in the expression,16 and $b^{\frac{1}{3}}$ , and both are raised to the power of $\frac{3}{2}$ .
$\left(16 b^{\frac{1}{3}}\right)^{\frac{3}{2}} =16^{\frac{3}{2}} \times \left(b^{\frac{1}{3}}\right)^{\frac{3}{2}}$

STEP 4

Now, calculate the power of16. $16^{\frac{3}{2}} = \sqrt{16^3}$

STEP 5

Calculate the value of $16^3$ .
$16^3 =409$

STEP 6

Now, calculate the square root of4096.
$\sqrt{4096} =64$

STEP 7

Now, calculate the power of $b^{\frac{1}{3}}$ .
$\left(b^{\frac{1}{3}}\right)^{\frac{3}{2}} = b^{\frac{1}{3} \times \frac{3}{2}}$

STEP 8

Multiply the exponents.
$b^{\frac{1}{3} \times \frac{3}{2}} = b^{\frac{1}{2}}$

STEP 9

Now, combine the results from steps6 and8 to write the original expression as a power.
$\left(16 b^{\frac{}{3}}\right)^{\frac{3}{2}} =64b^{\frac{}{2}}$ So, $\left(16 b^{\frac{}{3}}\right)^{\frac{3}{2}}$ as a power is $64b^{\frac{}{2}}$ .

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Simplify the expression (16b13)32\left(16 b^{\frac{1}{3}}\right)^{\frac{3}{2}}(16b31​)23​ as a power of bbb.

STEP 1

Assumptions1. We are given the expression (16b13)3\left(16 b^{\frac{1}{3}}\right)^{\frac{3}{}}(16b31​)3​. . We want to write this expression as a power.

STEP 2

First, we need to understand that when we raise a power to another power, we multiply the exponents. This is a basic rule of exponents.(am)n=am×n\left(a^m\right)^n = a^{m \times n}(am)n=am×n

STEP 3

STEP 4

Now, calculate the power of16.1632=16316^{\frac{3}{2}} = \sqrt{16^3}1623​=163​

STEP 5

Calculate the value of 16316^3163.163=40916^3 =409163=409

STEP 6

Now, calculate the square root of4096.4096=64\sqrt{4096} =644096​=64

STEP 7

Now, calculate the power of b13b^{\frac{1}{3}}b31​.(b13)32=b13×32\left(b^{\frac{1}{3}}\right)^{\frac{3}{2}} = b^{\frac{1}{3} \times \frac{3}{2}}(b31​)23​=b31​×23​

STEP 8

Multiply the exponents.b13×32=b12b^{\frac{1}{3} \times \frac{3}{2}} = b^{\frac{1}{2}}b31​×23​=b21​

STEP 9

Now, combine the results from steps6 and8 to write the original expression as a power.(16b3)32=64b2\left(16 b^{\frac{}{3}}\right)^{\frac{3}{2}} =64b^{\frac{}{2}}(16b3​)23​=64b2​So, (16b3)32\left(16 b^{\frac{}{3}}\right)^{\frac{3}{2}}(16b3​)23​ as a power is 64b264b^{\frac{}{2}}64b2​.

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Simplify the expression $\left(16 b^{\frac{1}{3}}\right)^{\frac{3}{2}}$ as a power of $b$ .

Assumptions1. We are given the expression $\left(16 b^{\frac{1}{3}}\right)^{\frac{3}{}}$ . . We want to write this expression as a power.

First, we need to understand that when we raise a power to another power, we multiply the exponents. This is a basic rule of exponents. $\left(a^m\right)^n = a^{m \times n}$

Now, calculate the power of16. $16^{\frac{3}{2}} = \sqrt{16^3}$

Calculate the value of $16^3$ .
$16^3 =409$

Now, calculate the square root of4096.
$\sqrt{4096} =64$

Now, calculate the power of $b^{\frac{1}{3}}$ .
$\left(b^{\frac{1}{3}}\right)^{\frac{3}{2}} = b^{\frac{1}{3} \times \frac{3}{2}}$

Multiply the exponents.
$b^{\frac{1}{3} \times \frac{3}{2}} = b^{\frac{1}{2}}$

Now, combine the results from steps6 and8 to write the original expression as a power.
$\left(16 b^{\frac{}{3}}\right)^{\frac{3}{2}} =64b^{\frac{}{2}}$ So, $\left(16 b^{\frac{}{3}}\right)^{\frac{3}{2}}$ as a power is $64b^{\frac{}{2}}$ .