Solved on Dec 07, 2023

Write the expression using radical notation and simplify if possible, assuming all variables represent nonnegative quantities. $\sqrt{64 x^{4}}$

STEP 1

Assumptions
1. The expression is $\left(64 x^{4}\right)^{\frac{1}{2}}$ .
2. All variables represent nonnegative quantities.
3. Radical notation means expressing the quantity using a root, such as a square root, cube root, etc.

STEP 2

Understand the exponent rule that states $(a^m)^n = a^{mn}$ , where $a$ is a real number and $m$ and $n$ are integers. This rule will be used to simplify the expression.

STEP 3

Apply the exponent rule to the given expression.
$\left(64 x^{4}\right)^{\frac{1}{2}} = 64^{\frac{1}{2}} \cdot \left(x^{4}\right)^{\frac{1}{2}}$

STEP 4

Simplify the expression by finding the square root of $64$ and applying the exponent rule to $x^4$ .
$64^{\frac{1}{2}} = \sqrt{64}$
$\left(x^{4}\right)^{\frac{1}{2}} = x^{4 \cdot \frac{1}{2}}$

STEP 5

Calculate the square root of $64$ .
$\sqrt{64} = 8$

STEP 6

Simplify the exponent for $x$ .
$x^{4 \cdot \frac{1}{2}} = x^{2}$

STEP 7

Combine the simplified terms to write the final expression in radical notation.
$\left(64 x^{4}\right)^{\frac{1}{2}} = 8x^{2}$
The expression $\left(64 x^{4}\right)^{\frac{1}{2}}$ in radical notation and simplified form is $8x^{2}$ .

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Write the expression using radical notation and simplify if possible, assuming all variables represent nonnegative quantities. $\sqrt{64 x^{4}}$

STEP 1

Assumptions
1. The expression is $\left(64 x^{4}\right)^{\frac{1}{2}}$ .
2. All variables represent nonnegative quantities.
3. Radical notation means expressing the quantity using a root, such as a square root, cube root, etc.

STEP 2

Understand the exponent rule that states $(a^m)^n = a^{mn}$ , where $a$ is a real number and $m$ and $n$ are integers. This rule will be used to simplify the expression.

STEP 3

Apply the exponent rule to the given expression.
$\left(64 x^{4}\right)^{\frac{1}{2}} = 64^{\frac{1}{2}} \cdot \left(x^{4}\right)^{\frac{1}{2}}$

STEP 4

Simplify the expression by finding the square root of $64$ and applying the exponent rule to $x^4$ .
$64^{\frac{1}{2}} = \sqrt{64}$
$\left(x^{4}\right)^{\frac{1}{2}} = x^{4 \cdot \frac{1}{2}}$

STEP 5

Calculate the square root of $64$ .
$\sqrt{64} = 8$

STEP 6

Simplify the exponent for $x$ .
$x^{4 \cdot \frac{1}{2}} = x^{2}$

STEP 7

Combine the simplified terms to write the final expression in radical notation.
$\left(64 x^{4}\right)^{\frac{1}{2}} = 8x^{2}$
The expression $\left(64 x^{4}\right)^{\frac{1}{2}}$ in radical notation and simplified form is $8x^{2}$ .

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Write the expression using radical notation and simplify if possible, assuming all variables represent nonnegative quantities. 64x4\sqrt{64 x^{4}}64x4​

STEP 1

Assumptions1. The expression is (64x4)12\left(64 x^{4}\right)^{\frac{1}{2}}(64x4)21​.2. All variables represent nonnegative quantities.3. Radical notation means expressing the quantity using a root, such as a square root, cube root, etc.

STEP 2

Understand the exponent rule that states (am)n=amn(a^m)^n = a^{mn}(am)n=amn, where aaa is a real number and mmm and nnn are integers. This rule will be used to simplify the expression.

STEP 3

Apply the exponent rule to the given expression.(64x4)12=6412⋅(x4)12 \left(64 x^{4}\right)^{\frac{1}{2}} = 64^{\frac{1}{2}} \cdot \left(x^{4}\right)^{\frac{1}{2}} (64x4)21​=6421​⋅(x4)21​

STEP 4

Simplify the expression by finding the square root of 646464 and applying the exponent rule to x4x^4x4.6412=64 64^{\frac{1}{2}} = \sqrt{64} 6421​=64​(x4)12=x4⋅12 \left(x^{4}\right)^{\frac{1}{2}} = x^{4 \cdot \frac{1}{2}} (x4)21​=x4⋅21​

STEP 5

Calculate the square root of 646464.64=8 \sqrt{64} = 8 64​=8

STEP 6

Simplify the exponent for xxx.x4⋅12=x2 x^{4 \cdot \frac{1}{2}} = x^{2} x4⋅21​=x2

STEP 7

Combine the simplified terms to write the final expression in radical notation.(64x4)12=8x2 \left(64 x^{4}\right)^{\frac{1}{2}} = 8x^{2} (64x4)21​=8x2The expression (64x4)12\left(64 x^{4}\right)^{\frac{1}{2}}(64x4)21​ in radical notation and simplified form is 8x28x^{2}8x2.

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Write the expression using radical notation and simplify if possible, assuming all variables represent nonnegative quantities. $\sqrt{64 x^{4}}$

Assumptions
1. The expression is $\left(64 x^{4}\right)^{\frac{1}{2}}$ .
2. All variables represent nonnegative quantities.
3. Radical notation means expressing the quantity using a root, such as a square root, cube root, etc.

Understand the exponent rule that states $(a^m)^n = a^{mn}$ , where $a$ is a real number and $m$ and $n$ are integers. This rule will be used to simplify the expression.

Apply the exponent rule to the given expression.
$\left(64 x^{4}\right)^{\frac{1}{2}} = 64^{\frac{1}{2}} \cdot \left(x^{4}\right)^{\frac{1}{2}}$

Simplify the expression by finding the square root of $64$ and applying the exponent rule to $x^4$ .
$64^{\frac{1}{2}} = \sqrt{64}$
$\left(x^{4}\right)^{\frac{1}{2}} = x^{4 \cdot \frac{1}{2}}$

Calculate the square root of $64$ .
$\sqrt{64} = 8$

Simplify the exponent for $x$ .
$x^{4 \cdot \frac{1}{2}} = x^{2}$

Combine the simplified terms to write the final expression in radical notation.
$\left(64 x^{4}\right)^{\frac{1}{2}} = 8x^{2}$
The expression $\left(64 x^{4}\right)^{\frac{1}{2}}$ in radical notation and simplified form is $8x^{2}$ .