Algebra Examples

$(2 x^{9} y^{n}) (4 x^{2} y^{10}) = 8 x^{11} y^{20}$

Step 1

Simplify.

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Step 1.1

Multiply $4$ by $2$ .

$8 x^{9} y^{n} (x^{2} y^{10}) = 8 x^{11} y^{20}$

Step 1.2

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

$8 x^{2 + 9} y^{n} y^{10} = 8 x^{11} y^{20}$

Step 1.3

Add $2$ and $9$ .

$8 x^{11} y^{n} y^{10} = 8 x^{11} y^{20}$

Step 1.4

Multiply $y^{n}$ by $y^{10}$ by adding the exponents.

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Step 1.4.1

Move $y^{10}$ .

$8 x^{11} (y^{10} y^{n}) = 8 x^{11} y^{20}$

Step 1.4.2

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

$8 x^{11} y^{10 + n} = 8 x^{11} y^{20}$

Step 2

Divide each term in $8 x^{11} y^{10 + n} = 8 x^{11} y^{20}$ by $8 x^{11}$ and simplify.

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Step 2.1

Divide each term in $8 x^{11} y^{10 + n} = 8 x^{11} y^{20}$ by $8 x^{11}$ .

$\frac{8 x^{11} y^{10 + n}}{8 x^{11}} = \frac{8 x^{11} y^{20}}{8 x^{11}}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $8$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{8 x^{11} y^{10 + n}}{8 x^{11}} = \frac{8 x^{11} y^{20}}{8 x^{11}}$

Step 2.2.1.2

Rewrite the expression.

$\frac{x^{11} y^{10 + n}}{x^{11}} = \frac{8 x^{11} y^{20}}{8 x^{11}}$

Step 2.2.2

Cancel the common factor of $x^{11}$ .

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Step 2.2.2.1

Cancel the common factor.

$\frac{x^{11} y^{10 + n}}{x^{11}} = \frac{8 x^{11} y^{20}}{8 x^{11}}$

Step 2.2.2.2

Divide $y^{10 + n}$ by $1$ .

$y^{10 + n} = \frac{8 x^{11} y^{20}}{8 x^{11}}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $8$ .

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Step 2.3.1.1

Cancel the common factor.

$y^{10 + n} = \frac{8 x^{11} y^{20}}{8 x^{11}}$

Step 2.3.1.2

Rewrite the expression.

$y^{10 + n} = \frac{x^{11} y^{20}}{x^{11}}$

Step 2.3.2

Cancel the common factor of $x^{11}$ .

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Step 2.3.2.1

Cancel the common factor.

$y^{10 + n} = \frac{x^{11} y^{20}}{x^{11}}$

Step 2.3.2.2

Divide $y^{20}$ by $1$ .

$y^{10 + n} = y^{20}$

Step 3

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$10 + n = 20$

Step 4

Move all terms not containing $n$ to the right side of the equation.

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Step 4.1

Subtract $10$ from both sides of the equation.

$n = 20 - 10$

Step 4.2

Subtract $10$ from $20$ .

$n = 10$