Algebra Examples

$\frac{{(2 x^{n + 1})}^{2} x^{3 - n}}{x^{2 (n + 1)} {(x^{n})}^{2}}$

Step 1

Cancel the common factor of $x^{3 - n}$ and $x^{2 (n + 1)}$ .

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Step 1.1

Factor $x^{2 (n + 1)}$ out of ${(2 x^{n + 1})}^{2} x^{3 - n}$ .

$\frac{x^{2 (n + 1)} ({(2 x^{n + 1})}^{2} x^{3 - n - 2 (n + 1)})}{x^{2 (n + 1)} {(x^{n})}^{2}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $x^{2 (n + 1)}$ out of $x^{2 (n + 1)} {(x^{n})}^{2}$ .

$\frac{x^{2 (n + 1)} ({(2 x^{n + 1})}^{2} x^{3 - n - 2 (n + 1)})}{x^{2 (n + 1)} ({(x^{n})}^{2})}$

Step 1.2.2

Cancel the common factor.

$\frac{x^{2 (n + 1)} ({(2 x^{n + 1})}^{2} x^{3 - n - 2 (n + 1)})}{x^{2 (n + 1)} {(x^{n})}^{2}}$

Step 1.2.3

Rewrite the expression.

$\frac{{(2 x^{n + 1})}^{2} x^{3 - n - 2 (n + 1)}}{{(x^{n})}^{2}}$

Step 2

Simplify the numerator.

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Step 2.1

Apply the product rule to $2 x^{n + 1}$ .

$\frac{2^{2} {(x^{n + 1})}^{2} x^{3 - n - 2 (n + 1)}}{{(x^{n})}^{2}}$

Step 2.2

Raise $2$ to the power of $2$ .

$\frac{4 {(x^{n + 1})}^{2} x^{3 - n - 2 (n + 1)}}{{(x^{n})}^{2}}$

Step 2.3

Multiply the exponents in ${(x^{n + 1})}^{2}$ .

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Step 2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{4 x^{(n + 1) \cdot 2} x^{3 - n - 2 (n + 1)}}{{(x^{n})}^{2}}$

Step 2.3.2

Apply the distributive property.

$\frac{4 x^{n \cdot 2 + 1 \cdot 2} x^{3 - n - 2 (n + 1)}}{{(x^{n})}^{2}}$

Step 2.3.3

Move $2$ to the left of $n$ .

$\frac{4 x^{2 \cdot n + 1 \cdot 2} x^{3 - n - 2 (n + 1)}}{{(x^{n})}^{2}}$

Step 2.3.4

Multiply $2$ by $1$ .

$\frac{4 x^{2 n + 2} x^{3 - n - 2 (n + 1)}}{{(x^{n})}^{2}}$

Step 2.4

Simplify each term.

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Step 2.4.1

Apply the distributive property.

$\frac{4 x^{2 n + 2} x^{3 - n - 2 n - 2 \cdot 1}}{{(x^{n})}^{2}}$

Step 2.4.2

Multiply $- 2$ by $1$ .

$\frac{4 x^{2 n + 2} x^{3 - n - 2 n - 2}}{{(x^{n})}^{2}}$

Step 2.5

Subtract $2$ from $3$ .

$\frac{4 x^{2 n + 2} x^{- n - 2 n + 1}}{{(x^{n})}^{2}}$

Step 2.6

Subtract $2 n$ from $- n$ .

$\frac{4 x^{2 n + 2} x^{- 3 n + 1}}{{(x^{n})}^{2}}$

Step 2.7

Multiply $x^{2 n + 2}$ by $x^{- 3 n + 1}$ by adding the exponents.

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Step 2.7.1

Move $x^{- 3 n + 1}$ .

$\frac{4 (x^{- 3 n + 1} x^{2 n + 2})}{{(x^{n})}^{2}}$

Step 2.7.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 x^{- 3 n + 1 + 2 n + 2}}{{(x^{n})}^{2}}$

Step 2.7.3

Add $- 3 n$ and $2 n$ .

$\frac{4 x^{- n + 1 + 2}}{{(x^{n})}^{2}}$

Step 2.7.4

Add $1$ and $2$ .

$\frac{4 x^{- n + 3}}{{(x^{n})}^{2}}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Multiply the exponents in ${(x^{n})}^{2}$ .

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Step 3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{4 x^{- n + 3}}{x^{n \cdot 2}}$

Step 3.1.2

Move $2$ to the left of $n$ .

$\frac{4 x^{- n + 3}}{x^{2 n}}$

Step 3.2

Cancel the common factor of $x^{- n + 3}$ and $x^{2 n}$ .

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Step 3.2.1

Factor $x^{2 n}$ out of $4 x^{- n + 3}$ .

$\frac{x^{2 n} (4 x^{- 3 n + 3})}{x^{2 n}}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Multiply by $1$ .

$\frac{x^{2 n} (4 x^{- 3 n + 3})}{x^{2 n} \cdot 1}$

Step 3.2.2.2

Cancel the common factor.

$\frac{x^{2 n} (4 x^{- 3 n + 3})}{x^{2 n} \cdot 1}$

Step 3.2.2.3

Rewrite the expression.

$\frac{4 x^{- 3 n + 3}}{1}$

Step 3.2.2.4

Divide $4 x^{- 3 n + 3}$ by $1$ .

$4 x^{- 3 n + 3}$