Math / Algebra

Question $\frac{2}{\frac{2}{3}}\left(\sqrt{\frac{x^{8}}{x^{3}}}\right)^{3}$

Studdy Solution

STEP 1

1. We need to simplify the given expression.
2. The expression involves division, square roots, and exponents.
3. We will use properties of exponents and radicals to simplify the expression.

STEP 2

1. Simplify the fraction $\frac{2}{\frac{2}{3}}$ .
2. Simplify the expression inside the square root $\sqrt{\frac{x^{8}}{x^{3}}}$ .
3. Simplify the entire expression by raising the simplified square root to the power of 3.

STEP 3

Simplify the fraction $\frac{2}{\frac{2}{3}}$ by multiplying by the reciprocal of the denominator:
$\frac{2}{\frac{2}{3}} = 2 \times \frac{3}{2} = 3$

STEP 4

Simplify the expression inside the square root $\sqrt{\frac{x^{8}}{x^{3}}}$ using the properties of exponents:
$\frac{x^{8}}{x^{3}} = x^{8-3} = x^{5}$

STEP 5

Simplify the square root $\sqrt{x^{5}}$ using the property $\sqrt{x^{n}} = x^{n/2}$ :
$\sqrt{x^{5}} = x^{5/2}$

STEP 6

Raise the simplified square root to the power of 3:
$(x^{5/2})^{3} = x^{(5/2) \times 3} = x^{15/2}$

STEP 7

Multiply the result from STEP_1 by the result from STEP_4:
$3 \times x^{15/2} = 3x^{15/2}$
This is the simplified form of the original expression.

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Question $\frac{2}{\frac{2}{3}}\left(\sqrt{\frac{x^{8}}{x^{3}}}\right)^{3}$

Studdy Solution

STEP 1

1. We need to simplify the given expression.
2. The expression involves division, square roots, and exponents.
3. We will use properties of exponents and radicals to simplify the expression.

STEP 2

1. Simplify the fraction $\frac{2}{\frac{2}{3}}$ .
2. Simplify the expression inside the square root $\sqrt{\frac{x^{8}}{x^{3}}}$ .
3. Simplify the entire expression by raising the simplified square root to the power of 3.

STEP 3

Simplify the fraction $\frac{2}{\frac{2}{3}}$ by multiplying by the reciprocal of the denominator:
$\frac{2}{\frac{2}{3}} = 2 \times \frac{3}{2} = 3$

STEP 4

Simplify the expression inside the square root $\sqrt{\frac{x^{8}}{x^{3}}}$ using the properties of exponents:
$\frac{x^{8}}{x^{3}} = x^{8-3} = x^{5}$

STEP 5

Simplify the square root $\sqrt{x^{5}}$ using the property $\sqrt{x^{n}} = x^{n/2}$ :
$\sqrt{x^{5}} = x^{5/2}$

STEP 6

Raise the simplified square root to the power of 3:
$(x^{5/2})^{3} = x^{(5/2) \times 3} = x^{15/2}$

STEP 7

Multiply the result from STEP_1 by the result from STEP_4:
$3 \times x^{15/2} = 3x^{15/2}$
This is the simplified form of the original expression.

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Question223(x8x3)3\frac{2}{\frac{2}{3}}\left(\sqrt{\frac{x^{8}}{x^{3}}}\right)^{3}32​2​(x3x8​​)3

Studdy Solution

STEP 1

1. We need to simplify the given expression.2. The expression involves division, square roots, and exponents.3. We will use properties of exponents and radicals to simplify the expression.

STEP 2

1. Simplify the fraction 223\frac{2}{\frac{2}{3}}32​2​.2. Simplify the expression inside the square root x8x3\sqrt{\frac{x^{8}}{x^{3}}}x3x8​​.3. Simplify the entire expression by raising the simplified square root to the power of 3.

STEP 3

Simplify the fraction 223\frac{2}{\frac{2}{3}}32​2​ by multiplying by the reciprocal of the denominator:223=2×32=3\frac{2}{\frac{2}{3}} = 2 \times \frac{3}{2} = 332​2​=2×23​=3

STEP 4

Simplify the expression inside the square root x8x3\sqrt{\frac{x^{8}}{x^{3}}}x3x8​​ using the properties of exponents:x8x3=x8−3=x5\frac{x^{8}}{x^{3}} = x^{8-3} = x^{5}x3x8​=x8−3=x5

STEP 5

Simplify the square root x5\sqrt{x^{5}}x5​ using the property xn=xn/2\sqrt{x^{n}} = x^{n/2}xn​=xn/2:x5=x5/2\sqrt{x^{5}} = x^{5/2}x5​=x5/2

STEP 6

Raise the simplified square root to the power of 3:(x5/2)3=x(5/2)×3=x15/2(x^{5/2})^{3} = x^{(5/2) \times 3} = x^{15/2}(x5/2)3=x(5/2)×3=x15/2

STEP 7

Multiply the result from STEP_1 by the result from STEP_4:3×x15/2=3x15/23 \times x^{15/2} = 3x^{15/2}3×x15/2=3x15/2This is the simplified form of the original expression.

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Question $\frac{2}{\frac{2}{3}}\left(\sqrt{\frac{x^{8}}{x^{3}}}\right)^{3}$

1. We need to simplify the given expression.
2. The expression involves division, square roots, and exponents.
3. We will use properties of exponents and radicals to simplify the expression.

1. Simplify the fraction $\frac{2}{\frac{2}{3}}$ .
2. Simplify the expression inside the square root $\sqrt{\frac{x^{8}}{x^{3}}}$ .
3. Simplify the entire expression by raising the simplified square root to the power of 3.

Simplify the fraction $\frac{2}{\frac{2}{3}}$ by multiplying by the reciprocal of the denominator:
$\frac{2}{\frac{2}{3}} = 2 \times \frac{3}{2} = 3$

Simplify the expression inside the square root $\sqrt{\frac{x^{8}}{x^{3}}}$ using the properties of exponents:
$\frac{x^{8}}{x^{3}} = x^{8-3} = x^{5}$

Simplify the square root $\sqrt{x^{5}}$ using the property $\sqrt{x^{n}} = x^{n/2}$ :
$\sqrt{x^{5}} = x^{5/2}$

Raise the simplified square root to the power of 3:
$(x^{5/2})^{3} = x^{(5/2) \times 3} = x^{15/2}$

Multiply the result from STEP_1 by the result from STEP_4:
$3 \times x^{15/2} = 3x^{15/2}$
This is the simplified form of the original expression.