Algebra Examples

$x^{k} y^{4} (2 x^{3} + 7 x^{2} y^{4}) = 2 x^{4} y^{4} + 7 x^{3} y^{8}$

Step 1

Divide each term in $x^{k} y^{4} (2 x^{3} + 7 x^{2} y^{4}) = 2 x^{4} y^{4} + 7 x^{3} y^{8}$ by $y^{4} (2 x^{3} + 7 x^{2} y^{4})$ and simplify.

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Step 1.1

Divide each term in $x^{k} y^{4} (2 x^{3} + 7 x^{2} y^{4}) = 2 x^{4} y^{4} + 7 x^{3} y^{8}$ by $y^{4} (2 x^{3} + 7 x^{2} y^{4})$ .

$\frac{x^{k} y^{4} (2 x^{3} + 7 x^{2} y^{4})}{y^{4} (2 x^{3} + 7 x^{2} y^{4})} = \frac{2 x^{4} y^{4}}{y^{4} (2 x^{3} + 7 x^{2} y^{4})} + \frac{7 x^{3} y^{8}}{y^{4} (2 x^{3} + 7 x^{2} y^{4})}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $y^{4}$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{x^{k} y^{4} (2 x^{3} + 7 x^{2} y^{4})}{y^{4} (2 x^{3} + 7 x^{2} y^{4})} = \frac{2 x^{4} y^{4}}{y^{4} (2 x^{3} + 7 x^{2} y^{4})} + \frac{7 x^{3} y^{8}}{y^{4} (2 x^{3} + 7 x^{2} y^{4})}$

Step 1.2.1.2

Rewrite the expression.

$\frac{x^{k} (2 x^{3} + 7 x^{2} y^{4})}{2 x^{3} + 7 x^{2} y^{4}} = \frac{2 x^{4} y^{4}}{y^{4} (2 x^{3} + 7 x^{2} y^{4})} + \frac{7 x^{3} y^{8}}{y^{4} (2 x^{3} + 7 x^{2} y^{4})}$

Step 1.2.2

Cancel the common factor of $2 x^{3} + 7 x^{2} y^{4}$ .

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Step 1.2.2.1

Cancel the common factor.

$\frac{x^{k} (2 x^{3} + 7 x^{2} y^{4})}{2 x^{3} + 7 x^{2} y^{4}} = \frac{2 x^{4} y^{4}}{y^{4} (2 x^{3} + 7 x^{2} y^{4})} + \frac{7 x^{3} y^{8}}{y^{4} (2 x^{3} + 7 x^{2} y^{4})}$

Step 1.2.2.2

Divide $x^{k}$ by $1$ .

$x^{k} = \frac{2 x^{4} y^{4}}{y^{4} (2 x^{3} + 7 x^{2} y^{4})} + \frac{7 x^{3} y^{8}}{y^{4} (2 x^{3} + 7 x^{2} y^{4})}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Cancel the common factor of $y^{4}$ .

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Step 1.3.1.1.1

Cancel the common factor.

$x^{k} = \frac{2 x^{4} y^{4}}{y^{4} (2 x^{3} + 7 x^{2} y^{4})} + \frac{7 x^{3} y^{8}}{y^{4} (2 x^{3} + 7 x^{2} y^{4})}$

Step 1.3.1.1.2

Rewrite the expression.

$x^{k} = \frac{2 x^{4}}{2 x^{3} + 7 x^{2} y^{4}} + \frac{7 x^{3} y^{8}}{y^{4} (2 x^{3} + 7 x^{2} y^{4})}$

Step 1.3.1.2

Factor $x^{2}$ out of $2 x^{3} + 7 x^{2} y^{4}$ .

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Step 1.3.1.2.1

Factor $x^{2}$ out of $2 x^{3}$ .

$x^{k} = \frac{2 x^{4}}{x^{2} (2 x) + 7 x^{2} y^{4}} + \frac{7 x^{3} y^{8}}{y^{4} (2 x^{3} + 7 x^{2} y^{4})}$

Step 1.3.1.2.2

Factor $x^{2}$ out of $7 x^{2} y^{4}$ .

$x^{k} = \frac{2 x^{4}}{x^{2} (2 x) + x^{2} (7 y^{4})} + \frac{7 x^{3} y^{8}}{y^{4} (2 x^{3} + 7 x^{2} y^{4})}$

Step 1.3.1.2.3

Factor $x^{2}$ out of $x^{2} (2 x) + x^{2} (7 y^{4})$ .

$x^{k} = \frac{2 x^{4}}{x^{2} (2 x + 7 y^{4})} + \frac{7 x^{3} y^{8}}{y^{4} (2 x^{3} + 7 x^{2} y^{4})}$

Step 1.3.1.3

Cancel the common factor of $x^{4}$ and $x^{2}$ .

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Step 1.3.1.3.1

Factor $x^{2}$ out of $2 x^{4}$ .

$x^{k} = \frac{x^{2} (2 x^{2})}{x^{2} (2 x + 7 y^{4})} + \frac{7 x^{3} y^{8}}{y^{4} (2 x^{3} + 7 x^{2} y^{4})}$

Step 1.3.1.3.2

Cancel the common factors.

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Step 1.3.1.3.2.1

Cancel the common factor.

$x^{k} = \frac{x^{2} (2 x^{2})}{x^{2} (2 x + 7 y^{4})} + \frac{7 x^{3} y^{8}}{y^{4} (2 x^{3} + 7 x^{2} y^{4})}$

Step 1.3.1.3.2.2

Rewrite the expression.

$x^{k} = \frac{2 x^{2}}{2 x + 7 y^{4}} + \frac{7 x^{3} y^{8}}{y^{4} (2 x^{3} + 7 x^{2} y^{4})}$

Step 1.3.1.4

Cancel the common factor of $y^{8}$ and $y^{4}$ .

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Step 1.3.1.4.1

Factor $y^{4}$ out of $7 x^{3} y^{8}$ .

$x^{k} = \frac{2 x^{2}}{2 x + 7 y^{4}} + \frac{y^{4} (7 x^{3} y^{4})}{y^{4} (2 x^{3} + 7 x^{2} y^{4})}$

Step 1.3.1.4.2

Cancel the common factors.

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Step 1.3.1.4.2.1

Cancel the common factor.

$x^{k} = \frac{2 x^{2}}{2 x + 7 y^{4}} + \frac{y^{4} (7 x^{3} y^{4})}{y^{4} (2 x^{3} + 7 x^{2} y^{4})}$

Step 1.3.1.4.2.2

Rewrite the expression.

$x^{k} = \frac{2 x^{2}}{2 x + 7 y^{4}} + \frac{7 x^{3} y^{4}}{2 x^{3} + 7 x^{2} y^{4}}$

Step 1.3.1.5

Factor $x^{2}$ out of $2 x^{3} + 7 x^{2} y^{4}$ .

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Step 1.3.1.5.1

Factor $x^{2}$ out of $2 x^{3}$ .

$x^{k} = \frac{2 x^{2}}{2 x + 7 y^{4}} + \frac{7 x^{3} y^{4}}{x^{2} (2 x) + 7 x^{2} y^{4}}$

Step 1.3.1.5.2

Factor $x^{2}$ out of $7 x^{2} y^{4}$ .

$x^{k} = \frac{2 x^{2}}{2 x + 7 y^{4}} + \frac{7 x^{3} y^{4}}{x^{2} (2 x) + x^{2} (7 y^{4})}$

Step 1.3.1.5.3

Factor $x^{2}$ out of $x^{2} (2 x) + x^{2} (7 y^{4})$ .

$x^{k} = \frac{2 x^{2}}{2 x + 7 y^{4}} + \frac{7 x^{3} y^{4}}{x^{2} (2 x + 7 y^{4})}$

Step 1.3.1.6

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

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Step 1.3.1.6.1

Factor $x^{2}$ out of $7 x^{3} y^{4}$ .

$x^{k} = \frac{2 x^{2}}{2 x + 7 y^{4}} + \frac{x^{2} (7 x y^{4})}{x^{2} (2 x + 7 y^{4})}$

Step 1.3.1.6.2

Cancel the common factors.

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Step 1.3.1.6.2.1

Cancel the common factor.

$x^{k} = \frac{2 x^{2}}{2 x + 7 y^{4}} + \frac{x^{2} (7 x y^{4})}{x^{2} (2 x + 7 y^{4})}$

Step 1.3.1.6.2.2

Rewrite the expression.

$x^{k} = \frac{2 x^{2}}{2 x + 7 y^{4}} + \frac{7 x y^{4}}{2 x + 7 y^{4}}$

Step 1.3.2

Simplify terms.

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Step 1.3.2.1

Combine the numerators over the common denominator.

$x^{k} = \frac{2 x^{2} + 7 x y^{4}}{2 x + 7 y^{4}}$

Step 1.3.2.2

Factor $x$ out of $2 x^{2} + 7 x y^{4}$ .

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Step 1.3.2.2.1

Factor $x$ out of $2 x^{2}$ .

$x^{k} = \frac{x (2 x) + 7 x y^{4}}{2 x + 7 y^{4}}$

Step 1.3.2.2.2

Factor $x$ out of $7 x y^{4}$ .

$x^{k} = \frac{x (2 x) + x (7 y^{4})}{2 x + 7 y^{4}}$

Step 1.3.2.2.3

Factor $x$ out of $x (2 x) + x (7 y^{4})$ .

$x^{k} = \frac{x (2 x + 7 y^{4})}{2 x + 7 y^{4}}$

Step 1.3.2.3

Cancel the common factor of $2 x + 7 y^{4}$ .

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Step 1.3.2.3.1

Cancel the common factor.

$x^{k} = \frac{x (2 x + 7 y^{4})}{2 x + 7 y^{4}}$

Step 1.3.2.3.2

Divide $x$ by $1$ .

$x^{k} = x$

Step 2

Create equivalent expressions in the equation that all have equal bases.

$x^{k} = x^{1}$

Step 3

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

$k = 1$