Solved on Jan 23, 2024

Simplify the square root of $(3/4)^9$ as a power.

STEP 1

Assumptions
1. We are given the expression $\sqrt{\left(\frac{3}{4}\right)^{9}}$ .
2. We need to write the expression as a power.

STEP 2

First, we need to understand the property of square roots and powers. The square root of a number is the same as raising that number to the power of $\frac{1}{2}$ .
$\sqrt{a} = a^{\frac{1}{2}}$

STEP 3

Apply this property to the given expression.
$\sqrt{\left(\frac{3}{4}\right)^{9}} = \left(\left(\frac{3}{4}\right)^{9}\right)^{\frac{1}{2}}$

STEP 4

When raising a power to another power, we multiply the exponents according to the laws of exponents.
$\left(a^{m}\right)^{n} = a^{m \cdot n}$

STEP 5

Apply this law to the expression.
$\left(\left(\frac{3}{4}\right)^{9}\right)^{\frac{1}{2}} = \left(\frac{3}{4}\right)^{9 \cdot \frac{1}{2}}$

STEP 6

Multiply the exponents.
$9 \cdot \frac{1}{2} = \frac{9}{2}$

STEP 7

Write the final expression as a power.
$\left(\frac{3}{4}\right)^{9 \cdot \frac{1}{2}} = \left(\frac{3}{4}\right)^{\frac{9}{2}}$
The expression $\sqrt{\left(\frac{3}{4}\right)^{9}}$ written as a power is $\left(\frac{3}{4}\right)^{\frac{9}{2}}$ .

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Simplify the square root of $(3/4)^9$ as a power.

STEP 1

Assumptions
1. We are given the expression $\sqrt{\left(\frac{3}{4}\right)^{9}}$ .
2. We need to write the expression as a power.

STEP 2

First, we need to understand the property of square roots and powers. The square root of a number is the same as raising that number to the power of $\frac{1}{2}$ .
$\sqrt{a} = a^{\frac{1}{2}}$

STEP 3

Apply this property to the given expression.
$\sqrt{\left(\frac{3}{4}\right)^{9}} = \left(\left(\frac{3}{4}\right)^{9}\right)^{\frac{1}{2}}$

STEP 4

When raising a power to another power, we multiply the exponents according to the laws of exponents.
$\left(a^{m}\right)^{n} = a^{m \cdot n}$

STEP 5

Apply this law to the expression.
$\left(\left(\frac{3}{4}\right)^{9}\right)^{\frac{1}{2}} = \left(\frac{3}{4}\right)^{9 \cdot \frac{1}{2}}$

STEP 6

Multiply the exponents.
$9 \cdot \frac{1}{2} = \frac{9}{2}$

STEP 7

Write the final expression as a power.
$\left(\frac{3}{4}\right)^{9 \cdot \frac{1}{2}} = \left(\frac{3}{4}\right)^{\frac{9}{2}}$
The expression $\sqrt{\left(\frac{3}{4}\right)^{9}}$ written as a power is $\left(\frac{3}{4}\right)^{\frac{9}{2}}$ .

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Simplify the square root of (3/4)9(3/4)^9(3/4)9 as a power.

STEP 1

Assumptions1. We are given the expression (34)9\sqrt{\left(\frac{3}{4}\right)^{9}}(43​)9​.2. We need to write the expression as a power.

STEP 2

First, we need to understand the property of square roots and powers. The square root of a number is the same as raising that number to the power of 12\frac{1}{2}21​.a=a12\sqrt{a} = a^{\frac{1}{2}}a​=a21​

STEP 3

Apply this property to the given expression.(34)9=((34)9)12\sqrt{\left(\frac{3}{4}\right)^{9}} = \left(\left(\frac{3}{4}\right)^{9}\right)^{\frac{1}{2}}(43​)9​=((43​)9)21​

STEP 4

When raising a power to another power, we multiply the exponents according to the laws of exponents.(am)n=am⋅n\left(a^{m}\right)^{n} = a^{m \cdot n}(am)n=am⋅n

STEP 5

Apply this law to the expression.((34)9)12=(34)9⋅12\left(\left(\frac{3}{4}\right)^{9}\right)^{\frac{1}{2}} = \left(\frac{3}{4}\right)^{9 \cdot \frac{1}{2}}((43​)9)21​=(43​)9⋅21​

STEP 6

Multiply the exponents.9⋅12=929 \cdot \frac{1}{2} = \frac{9}{2}9⋅21​=29​

STEP 7

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Simplify the square root of $(3/4)^9$ as a power.

Assumptions
1. We are given the expression $\sqrt{\left(\frac{3}{4}\right)^{9}}$ .
2. We need to write the expression as a power.

First, we need to understand the property of square roots and powers. The square root of a number is the same as raising that number to the power of $\frac{1}{2}$ .
$\sqrt{a} = a^{\frac{1}{2}}$

Apply this property to the given expression.
$\sqrt{\left(\frac{3}{4}\right)^{9}} = \left(\left(\frac{3}{4}\right)^{9}\right)^{\frac{1}{2}}$

When raising a power to another power, we multiply the exponents according to the laws of exponents.
$\left(a^{m}\right)^{n} = a^{m \cdot n}$

Apply this law to the expression.
$\left(\left(\frac{3}{4}\right)^{9}\right)^{\frac{1}{2}} = \left(\frac{3}{4}\right)^{9 \cdot \frac{1}{2}}$

Multiply the exponents.
$9 \cdot \frac{1}{2} = \frac{9}{2}$