Solved on Jan 22, 2024

Simplify the expression $\left(\frac{3}{8}\right)^{-\frac{1}{2}} \times (2 \sqrt{3})^{3}$ .

STEP 1

Assumptions
1. We need to simplify the expression $\left(\frac{3}{8}\right)^{-\frac{1}{2}} \times(2 \sqrt{3})^{3}$ .
2. We will use the properties of exponents to simplify the expression.

STEP 2

First, we will simplify the term $\left(\frac{3}{8}\right)^{-\frac{1}{2}}$ . The negative exponent means we take the reciprocal of the base and the exponent $\frac{1}{2}$ means we take the square root.
$\left(\frac{3}{8}\right)^{-\frac{1}{2}} = \left(\frac{8}{3}\right)^{\frac{1}{2}}$

STEP 3

Next, we simplify the square root of the fraction.
$\left(\frac{8}{3}\right)^{\frac{1}{2}} = \sqrt{\frac{8}{3}}$

STEP 4

We can simplify the square root of the numerator and the denominator separately.
$\sqrt{\frac{8}{3}} = \frac{\sqrt{8}}{\sqrt{3}}$

STEP 5

Simplify the square root of 8, which is 2 times the square root of 2.
$\frac{\sqrt{8}}{\sqrt{3}} = \frac{2\sqrt{2}}{\sqrt{3}}$

STEP 6

Now we will simplify the term $(2 \sqrt{3})^{3}$ . We will use the property of exponents that $(ab)^{n} = a^{n}b^{n}$ .
$(2 \sqrt{3})^{3} = 2^{3}(\sqrt{3})^{3}$

STEP 7

Calculate the value of $2^{3}$ and $(\sqrt{3})^{3}$ separately.
$2^{3} = 8$ $(\sqrt{3})^{3} = \sqrt{3} \times \sqrt{3} \times \sqrt{3}$

STEP 8

We know that $\sqrt{3} \times \sqrt{3} = 3$ , so we can simplify $(\sqrt{3})^{3}$ .
$(\sqrt{3})^{3} = 3 \times \sqrt{3}$

STEP 9

Now we can combine the results from steps 7 and 8.
$(2 \sqrt{3})^{3} = 8 \times 3 \times \sqrt{3}$

STEP 10

Calculate the product.
$8 \times 3 \times \sqrt{3} = 24\sqrt{3}$

STEP 11

Now we have simplified both terms separately. The original expression has become:
$\frac{2\sqrt{2}}{\sqrt{3}} \times 24\sqrt{3}$

STEP 12

We can simplify the expression further by multiplying the numerators and denominators.
$\frac{2\sqrt{2}}{\sqrt{3}} \times 24\sqrt{3} = \frac{2\sqrt{2} \times 24\sqrt{3}}{\sqrt{3} \times \sqrt{3}}$

STEP 13

Simplify the denominator $\sqrt{3} \times \sqrt{3}$ , which equals 3.
$\frac{2\sqrt{2} \times 24\sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{2\sqrt{2} \times 24\sqrt{3}}{3}$

STEP 14

Now we can cancel the $\sqrt{3}$ in the numerator with one in the denominator and simplify the fraction.
$\frac{2\sqrt{2} \times 24\sqrt{3}}{3} = \frac{2 \times 24\sqrt{2}}{3}$

STEP 15

Divide 24 by 3.
$\frac{2 \times 24\sqrt{2}}{3} = 2 \times 8\sqrt{2}$

STEP 16

Calculate the final product.
$2 \times 8\sqrt{2} = 16\sqrt{2}$
The simplified form of the expression $\left(\frac{3}{8}\right)^{-\frac{1}{2}} \times(2 \sqrt{3})^{3}$ is $16\sqrt{2}$ .

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Simplify the expression $\left(\frac{3}{8}\right)^{-\frac{1}{2}} \times (2 \sqrt{3})^{3}$ .

STEP 1

Assumptions
1. We need to simplify the expression $\left(\frac{3}{8}\right)^{-\frac{1}{2}} \times(2 \sqrt{3})^{3}$ .
2. We will use the properties of exponents to simplify the expression.

STEP 2

First, we will simplify the term $\left(\frac{3}{8}\right)^{-\frac{1}{2}}$ . The negative exponent means we take the reciprocal of the base and the exponent $\frac{1}{2}$ means we take the square root.
$\left(\frac{3}{8}\right)^{-\frac{1}{2}} = \left(\frac{8}{3}\right)^{\frac{1}{2}}$

STEP 3

Next, we simplify the square root of the fraction.
$\left(\frac{8}{3}\right)^{\frac{1}{2}} = \sqrt{\frac{8}{3}}$

STEP 4

We can simplify the square root of the numerator and the denominator separately.
$\sqrt{\frac{8}{3}} = \frac{\sqrt{8}}{\sqrt{3}}$

STEP 5

Simplify the square root of 8, which is 2 times the square root of 2.
$\frac{\sqrt{8}}{\sqrt{3}} = \frac{2\sqrt{2}}{\sqrt{3}}$

STEP 6

Now we will simplify the term $(2 \sqrt{3})^{3}$ . We will use the property of exponents that $(ab)^{n} = a^{n}b^{n}$ .
$(2 \sqrt{3})^{3} = 2^{3}(\sqrt{3})^{3}$

STEP 7

Calculate the value of $2^{3}$ and $(\sqrt{3})^{3}$ separately.
$2^{3} = 8$ $(\sqrt{3})^{3} = \sqrt{3} \times \sqrt{3} \times \sqrt{3}$

STEP 8

We know that $\sqrt{3} \times \sqrt{3} = 3$ , so we can simplify $(\sqrt{3})^{3}$ .
$(\sqrt{3})^{3} = 3 \times \sqrt{3}$

STEP 9

Now we can combine the results from steps 7 and 8.
$(2 \sqrt{3})^{3} = 8 \times 3 \times \sqrt{3}$

STEP 10

Calculate the product.
$8 \times 3 \times \sqrt{3} = 24\sqrt{3}$

STEP 11

Now we have simplified both terms separately. The original expression has become:
$\frac{2\sqrt{2}}{\sqrt{3}} \times 24\sqrt{3}$

STEP 12

We can simplify the expression further by multiplying the numerators and denominators.
$\frac{2\sqrt{2}}{\sqrt{3}} \times 24\sqrt{3} = \frac{2\sqrt{2} \times 24\sqrt{3}}{\sqrt{3} \times \sqrt{3}}$

STEP 13

Simplify the denominator $\sqrt{3} \times \sqrt{3}$ , which equals 3.
$\frac{2\sqrt{2} \times 24\sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{2\sqrt{2} \times 24\sqrt{3}}{3}$

STEP 14

Now we can cancel the $\sqrt{3}$ in the numerator with one in the denominator and simplify the fraction.
$\frac{2\sqrt{2} \times 24\sqrt{3}}{3} = \frac{2 \times 24\sqrt{2}}{3}$

STEP 15

Divide 24 by 3.
$\frac{2 \times 24\sqrt{2}}{3} = 2 \times 8\sqrt{2}$

STEP 16

Calculate the final product.
$2 \times 8\sqrt{2} = 16\sqrt{2}$
The simplified form of the expression $\left(\frac{3}{8}\right)^{-\frac{1}{2}} \times(2 \sqrt{3})^{3}$ is $16\sqrt{2}$ .

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Simplify the expression (38)−12×(23)3\left(\frac{3}{8}\right)^{-\frac{1}{2}} \times (2 \sqrt{3})^{3}(83​)−21​×(23​)3.

STEP 1

Assumptions1. We need to simplify the expression (38)−12×(23)3\left(\frac{3}{8}\right)^{-\frac{1}{2}} \times(2 \sqrt{3})^{3}(83​)−21​×(23​)3.2. We will use the properties of exponents to simplify the expression.

STEP 2

STEP 3

Next, we simplify the square root of the fraction.(83)12=83\left(\frac{8}{3}\right)^{\frac{1}{2}} = \sqrt{\frac{8}{3}}(38​)21​=38​​

STEP 4

We can simplify the square root of the numerator and the denominator separately.83=83\sqrt{\frac{8}{3}} = \frac{\sqrt{8}}{\sqrt{3}}38​​=3​8​​

STEP 5

Simplify the square root of 8, which is 2 times the square root of 2.83=223\frac{\sqrt{8}}{\sqrt{3}} = \frac{2\sqrt{2}}{\sqrt{3}}3​8​​=3​22​​

STEP 6

Now we will simplify the term (23)3(2 \sqrt{3})^{3}(23​)3. We will use the property of exponents that (ab)n=anbn(ab)^{n} = a^{n}b^{n}(ab)n=anbn.(23)3=23(3)3(2 \sqrt{3})^{3} = 2^{3}(\sqrt{3})^{3}(23​)3=23(3​)3

STEP 7

Calculate the value of 232^{3}23 and (3)3(\sqrt{3})^{3}(3​)3 separately.23=82^{3} = 823=8 (3)3=3×3×3(\sqrt{3})^{3} = \sqrt{3} \times \sqrt{3} \times \sqrt{3}(3​)3=3​×3​×3​

STEP 8

We know that 3×3=3\sqrt{3} \times \sqrt{3} = 33​×3​=3, so we can simplify (3)3(\sqrt{3})^{3}(3​)3.(3)3=3×3(\sqrt{3})^{3} = 3 \times \sqrt{3}(3​)3=3×3​

STEP 9

Now we can combine the results from steps 7 and 8.(23)3=8×3×3(2 \sqrt{3})^{3} = 8 \times 3 \times \sqrt{3}(23​)3=8×3×3​

STEP 10

Calculate the product.8×3×3=2438 \times 3 \times \sqrt{3} = 24\sqrt{3}8×3×3​=243​

STEP 11

Now we have simplified both terms separately. The original expression has become:223×243\frac{2\sqrt{2}}{\sqrt{3}} \times 24\sqrt{3}3​22​​×243​

STEP 12

We can simplify the expression further by multiplying the numerators and denominators.223×243=22×2433×3\frac{2\sqrt{2}}{\sqrt{3}} \times 24\sqrt{3} = \frac{2\sqrt{2} \times 24\sqrt{3}}{\sqrt{3} \times \sqrt{3}}3​22​​×243​=3​×3​22​×243​​

STEP 13

Simplify the denominator 3×3\sqrt{3} \times \sqrt{3}3​×3​, which equals 3.22×2433×3=22×2433\frac{2\sqrt{2} \times 24\sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{2\sqrt{2} \times 24\sqrt{3}}{3}3​×3​22​×243​​=322​×243​​

STEP 14

Now we can cancel the 3\sqrt{3}3​ in the numerator with one in the denominator and simplify the fraction.22×2433=2×2423\frac{2\sqrt{2} \times 24\sqrt{3}}{3} = \frac{2 \times 24\sqrt{2}}{3}322​×243​​=32×242​​

STEP 15

Divide 24 by 3.2×2423=2×82\frac{2 \times 24\sqrt{2}}{3} = 2 \times 8\sqrt{2}32×242​​=2×82​

STEP 16

Calculate the final product.2×82=1622 \times 8\sqrt{2} = 16\sqrt{2}2×82​=162​The simplified form of the expression (38)−12×(23)3\left(\frac{3}{8}\right)^{-\frac{1}{2}} \times(2 \sqrt{3})^{3}(83​)−21​×(23​)3 is 16216\sqrt{2}162​.

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Simplify the expression $\left(\frac{3}{8}\right)^{-\frac{1}{2}} \times (2 \sqrt{3})^{3}$ .

Assumptions
1. We need to simplify the expression $\left(\frac{3}{8}\right)^{-\frac{1}{2}} \times(2 \sqrt{3})^{3}$ .
2. We will use the properties of exponents to simplify the expression.

Next, we simplify the square root of the fraction.
$\left(\frac{8}{3}\right)^{\frac{1}{2}} = \sqrt{\frac{8}{3}}$

We can simplify the square root of the numerator and the denominator separately.
$\sqrt{\frac{8}{3}} = \frac{\sqrt{8}}{\sqrt{3}}$

Simplify the square root of 8, which is 2 times the square root of 2.
$\frac{\sqrt{8}}{\sqrt{3}} = \frac{2\sqrt{2}}{\sqrt{3}}$

Now we will simplify the term $(2 \sqrt{3})^{3}$ . We will use the property of exponents that $(ab)^{n} = a^{n}b^{n}$ .
$(2 \sqrt{3})^{3} = 2^{3}(\sqrt{3})^{3}$

Calculate the value of $2^{3}$ and $(\sqrt{3})^{3}$ separately.
$2^{3} = 8$ $(\sqrt{3})^{3} = \sqrt{3} \times \sqrt{3} \times \sqrt{3}$

We know that $\sqrt{3} \times \sqrt{3} = 3$ , so we can simplify $(\sqrt{3})^{3}$ .
$(\sqrt{3})^{3} = 3 \times \sqrt{3}$

Now we can combine the results from steps 7 and 8.
$(2 \sqrt{3})^{3} = 8 \times 3 \times \sqrt{3}$

Calculate the product.
$8 \times 3 \times \sqrt{3} = 24\sqrt{3}$

Now we have simplified both terms separately. The original expression has become:
$\frac{2\sqrt{2}}{\sqrt{3}} \times 24\sqrt{3}$

We can simplify the expression further by multiplying the numerators and denominators.
$\frac{2\sqrt{2}}{\sqrt{3}} \times 24\sqrt{3} = \frac{2\sqrt{2} \times 24\sqrt{3}}{\sqrt{3} \times \sqrt{3}}$

Simplify the denominator $\sqrt{3} \times \sqrt{3}$ , which equals 3.
$\frac{2\sqrt{2} \times 24\sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{2\sqrt{2} \times 24\sqrt{3}}{3}$

Now we can cancel the $\sqrt{3}$ in the numerator with one in the denominator and simplify the fraction.
$\frac{2\sqrt{2} \times 24\sqrt{3}}{3} = \frac{2 \times 24\sqrt{2}}{3}$

Divide 24 by 3.
$\frac{2 \times 24\sqrt{2}}{3} = 2 \times 8\sqrt{2}$

Calculate the final product.
$2 \times 8\sqrt{2} = 16\sqrt{2}$
The simplified form of the expression $\left(\frac{3}{8}\right)^{-\frac{1}{2}} \times(2 \sqrt{3})^{3}$ is $16\sqrt{2}$ .