QuestionSimplify the expression: $\frac{4 x}{\sqrt[3]{2 x^{2} y}} \cdot \frac{\sqrt[3]{4 x y^{2}}}{\sqrt[3]{4 x y^{2}}}$ .

Studdy Solution

STEP 1

Assumptions1. The given expression is $\frac{4 x}{\sqrt[3]{ x^{} y}} \cdot \frac{\sqrt[3]{4 x y^{}}}{\sqrt[3]{4 x y^{}}}$

STEP 2

We can simplify the expression by cancelling out the common terms in the numerator and the denominator.The given expression is equivalent to $\frac{4 x}{\sqrt[]{2 x^{2} y}} \cdot1$

STEP 3

Now, we can simplify the denominator $\sqrt[3]{2 x^{2} y}$ by writing it in the form of $x^{\frac{1}{3}}$ .
$\frac{ x}{\sqrt[3]{2 x^{2} y}} = \frac{ x}{(2 x^{2} y)^{\frac{1}{3}}}$

STEP 4

We can simplify the expression further by applying the power rule, which states that $(a^{m})^{n} = a^{m \cdot n}$ .
$\frac{4 x}{(2 x^{2} y)^{\frac{1}{3}}} = \frac{4 x}{2^{\frac{1}{3}} x^{\frac{2}{3}} y^{\frac{1}{3}}}$

STEP 5

Now, we can simplify the expression by cancelling out the common terms in the numerator and the denominator.
$\frac{4 x}{2^{\frac{1}{3}} x^{\frac{2}{3}} y^{\frac{1}{3}}} = \frac{4}{2^{\frac{1}{3}}} x^{1-\frac{2}{3}} y^{-\frac{1}{3}}$

STEP 6

implify the expression further by calculating the power of x and y.
$\frac{4}{2^{\frac{1}{3}}} x^{1-\frac{2}{3}} y^{-\frac{1}{3}} = \frac{4}{2^{\frac{1}{3}}} x^{\frac{1}{3}} y^{-\frac{1}{3}}$

STEP 7

Finally, we can write the expression in a more simplified form.
$\frac{4}{2^{\frac{1}{3}}} x^{\frac{1}{3}} y^{-\frac{1}{3}} =2^{\frac{4}{3}} x^{\frac{1}{3}} y^{-\frac{1}{3}}$ So, the simplified form of the given expression is $2^{\frac{4}{3}} x^{\frac{1}{3}} y^{-\frac{1}{3}}$ .

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QuestionSimplify the expression: $\frac{4 x}{\sqrt[3]{2 x^{2} y}} \cdot \frac{\sqrt[3]{4 x y^{2}}}{\sqrt[3]{4 x y^{2}}}$ .

Studdy Solution

STEP 1

Assumptions1. The given expression is $\frac{4 x}{\sqrt[3]{ x^{} y}} \cdot \frac{\sqrt[3]{4 x y^{}}}{\sqrt[3]{4 x y^{}}}$

STEP 2

We can simplify the expression by cancelling out the common terms in the numerator and the denominator.The given expression is equivalent to $\frac{4 x}{\sqrt[]{2 x^{2} y}} \cdot1$

STEP 3

Now, we can simplify the denominator $\sqrt[3]{2 x^{2} y}$ by writing it in the form of $x^{\frac{1}{3}}$ .
$\frac{ x}{\sqrt[3]{2 x^{2} y}} = \frac{ x}{(2 x^{2} y)^{\frac{1}{3}}}$

STEP 4

We can simplify the expression further by applying the power rule, which states that $(a^{m})^{n} = a^{m \cdot n}$ .
$\frac{4 x}{(2 x^{2} y)^{\frac{1}{3}}} = \frac{4 x}{2^{\frac{1}{3}} x^{\frac{2}{3}} y^{\frac{1}{3}}}$

STEP 5

Now, we can simplify the expression by cancelling out the common terms in the numerator and the denominator.
$\frac{4 x}{2^{\frac{1}{3}} x^{\frac{2}{3}} y^{\frac{1}{3}}} = \frac{4}{2^{\frac{1}{3}}} x^{1-\frac{2}{3}} y^{-\frac{1}{3}}$

STEP 6

implify the expression further by calculating the power of x and y.
$\frac{4}{2^{\frac{1}{3}}} x^{1-\frac{2}{3}} y^{-\frac{1}{3}} = \frac{4}{2^{\frac{1}{3}}} x^{\frac{1}{3}} y^{-\frac{1}{3}}$

STEP 7

Finally, we can write the expression in a more simplified form.
$\frac{4}{2^{\frac{1}{3}}} x^{\frac{1}{3}} y^{-\frac{1}{3}} =2^{\frac{4}{3}} x^{\frac{1}{3}} y^{-\frac{1}{3}}$ So, the simplified form of the given expression is $2^{\frac{4}{3}} x^{\frac{1}{3}} y^{-\frac{1}{3}}$ .

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QuestionSimplify the expression: 4x2x2y3⋅4xy234xy23\frac{4 x}{\sqrt[3]{2 x^{2} y}} \cdot \frac{\sqrt[3]{4 x y^{2}}}{\sqrt[3]{4 x y^{2}}}32x2y​4x​⋅34xy2​34xy2​​.

Studdy Solution

STEP 1

Assumptions1. The given expression is 4xxy3⋅4xy34xy3\frac{4 x}{\sqrt[3]{ x^{} y}} \cdot \frac{\sqrt[3]{4 x y^{}}}{\sqrt[3]{4 x y^{}}}3xy​4x​⋅34xy​34xy​​

STEP 2

We can simplify the expression by cancelling out the common terms in the numerator and the denominator.The given expression is equivalent to4x2x2y⋅1\frac{4 x}{\sqrt[]{2 x^{2} y}} \cdot12x2y​4x​⋅1

STEP 3

Now, we can simplify the denominator 2x2y3\sqrt[3]{2 x^{2} y}32x2y​ by writing it in the form of x13x^{\frac{1}{3}}x31​.x2x2y3=x(2x2y)13\frac{ x}{\sqrt[3]{2 x^{2} y}} = \frac{ x}{(2 x^{2} y)^{\frac{1}{3}}}32x2y​x​=(2x2y)31​x​

STEP 4

We can simplify the expression further by applying the power rule, which states that (am)n=am⋅n(a^{m})^{n} = a^{m \cdot n}(am)n=am⋅n.4x(2x2y)13=4x213x23y13\frac{4 x}{(2 x^{2} y)^{\frac{1}{3}}} = \frac{4 x}{2^{\frac{1}{3}} x^{\frac{2}{3}} y^{\frac{1}{3}}}(2x2y)31​4x​=231​x32​y31​4x​

STEP 5

STEP 6

implify the expression further by calculating the power of x and y.4213x1−23y−13=4213x13y−13\frac{4}{2^{\frac{1}{3}}} x^{1-\frac{2}{3}} y^{-\frac{1}{3}} = \frac{4}{2^{\frac{1}{3}}} x^{\frac{1}{3}} y^{-\frac{1}{3}}231​4​x1−32​y−31​=231​4​x31​y−31​

STEP 7

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QuestionSimplify the expression: $\frac{4 x}{\sqrt[3]{2 x^{2} y}} \cdot \frac{\sqrt[3]{4 x y^{2}}}{\sqrt[3]{4 x y^{2}}}$ .

Assumptions1. The given expression is $\frac{4 x}{\sqrt[3]{ x^{} y}} \cdot \frac{\sqrt[3]{4 x y^{}}}{\sqrt[3]{4 x y^{}}}$

We can simplify the expression by cancelling out the common terms in the numerator and the denominator.The given expression is equivalent to $\frac{4 x}{\sqrt[]{2 x^{2} y}} \cdot1$

Now, we can simplify the denominator $\sqrt[3]{2 x^{2} y}$ by writing it in the form of $x^{\frac{1}{3}}$ .
$\frac{ x}{\sqrt[3]{2 x^{2} y}} = \frac{ x}{(2 x^{2} y)^{\frac{1}{3}}}$

We can simplify the expression further by applying the power rule, which states that $(a^{m})^{n} = a^{m \cdot n}$ .
$\frac{4 x}{(2 x^{2} y)^{\frac{1}{3}}} = \frac{4 x}{2^{\frac{1}{3}} x^{\frac{2}{3}} y^{\frac{1}{3}}}$

implify the expression further by calculating the power of x and y.
$\frac{4}{2^{\frac{1}{3}}} x^{1-\frac{2}{3}} y^{-\frac{1}{3}} = \frac{4}{2^{\frac{1}{3}}} x^{\frac{1}{3}} y^{-\frac{1}{3}}$