Algebra Examples

$\frac{(5 - 5 \sqrt{x}) (5 + 5 \sqrt{x})}{(\sqrt{x} (x - 5) (5 + 5 \sqrt{x}))}$

Step 1

Cancel the common factor of $5 - 5 \sqrt{x}$ and $5 + 5 \sqrt{x}$ .

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Step 1.1

Factor $5$ out of $(5 - 5 \sqrt{x}) (5 + 5 \sqrt{x})$ .

$\frac{5 ((1 - \sqrt{x}) (5 + 5 \sqrt{x}))}{\sqrt{x} (x - 5) (5 + 5 \sqrt{x})}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $5$ out of $\sqrt{x} (x - 5) (5 + 5 \sqrt{x})$ .

$\frac{5 ((1 - \sqrt{x}) (5 + 5 \sqrt{x}))}{5 (\sqrt{x} (x - 5) (1 + \sqrt{x}))}$

Step 1.2.2

Cancel the common factor.

$\frac{5 ((1 - \sqrt{x}) (5 + 5 \sqrt{x}))}{5 (\sqrt{x} (x - 5) (1 + \sqrt{x}))}$

Step 1.2.3

Rewrite the expression.

$\frac{(1 - \sqrt{x}) (5 + 5 \sqrt{x})}{\sqrt{x} (x - 5) (1 + \sqrt{x})}$

Step 2

Factor $5$ out of $5 + 5 \sqrt{x}$ .

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Step 2.1

Factor $5$ out of $5$ .

$\frac{(1 - \sqrt{x}) (5 \cdot 1 + 5 \sqrt{x})}{\sqrt{x} (x - 5) (1 + \sqrt{x})}$

Step 2.2

Factor $5$ out of $5 \cdot 1 + 5 \sqrt{x}$ .

$\frac{(1 - \sqrt{x}) (5 (1 + \sqrt{x}))}{\sqrt{x} (x - 5) (1 + \sqrt{x})}$

$\frac{(1 - \sqrt{x}) \cdot 5 (1 + \sqrt{x})}{\sqrt{x} (x - 5) (1 + \sqrt{x})}$

Step 3

Cancel the common factor of $1 + \sqrt{x}$ .

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Step 3.1

Cancel the common factor.

$\frac{(1 - \sqrt{x}) \cdot 5 (1 + \sqrt{x})}{\sqrt{x} (x - 5) (1 + \sqrt{x})}$

Step 3.2

Rewrite the expression.

$\frac{(1 - \sqrt{x}) \cdot 5}{\sqrt{x} (x - 5)}$

Step 4

Move $5$ to the left of $1 - \sqrt{x}$ .

$\frac{5 \cdot (1 - \sqrt{x})}{\sqrt{x} (x - 5)}$

Step 5

Multiply $\frac{5 \cdot (1 - \sqrt{x})}{\sqrt{x} (x - 5)}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{5 \cdot (1 - \sqrt{x})}{\sqrt{x} (x - 5)} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 6

Simplify terms.

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Step 6.1

Combine and simplify the denominator.

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Step 6.1.1

Multiply $\frac{5 \cdot (1 - \sqrt{x})}{\sqrt{x} (x - 5)}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{5 \cdot (1 - \sqrt{x}) \sqrt{x}}{\sqrt{x} (x - 5) \sqrt{x}}$

Step 6.1.2

Move $\sqrt{x}$ .

$\frac{5 \cdot (1 - \sqrt{x}) \sqrt{x}}{(x - 5) (\sqrt{x} \sqrt{x})}$

Step 6.1.3

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{5 \cdot (1 - \sqrt{x}) \sqrt{x}}{(x - 5) ({\sqrt{x}}^{1} \sqrt{x})}$

Step 6.1.4

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{5 \cdot (1 - \sqrt{x}) \sqrt{x}}{(x - 5) ({\sqrt{x}}^{1} {\sqrt{x}}^{1})}$

Step 6.1.5

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

$\frac{5 \cdot (1 - \sqrt{x}) \sqrt{x}}{(x - 5) {\sqrt{x}}^{1 + 1}}$

Step 6.1.6

Add $1$ and $1$ .

$\frac{5 \cdot (1 - \sqrt{x}) \sqrt{x}}{(x - 5) {\sqrt{x}}^{2}}$

Step 6.1.7

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 6.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{5 \cdot (1 - \sqrt{x}) \sqrt{x}}{(x - 5) {(x^{\frac{1}{2}})}^{2}}$

Step 6.1.7.2

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

$\frac{5 \cdot (1 - \sqrt{x}) \sqrt{x}}{(x - 5) x^{\frac{1}{2} \cdot 2}}$

Step 6.1.7.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{5 \cdot (1 - \sqrt{x}) \sqrt{x}}{(x - 5) x^{\frac{2}{2}}}$

Step 6.1.7.4

Cancel the common factor of $2$ .

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Step 6.1.7.4.1

Cancel the common factor.

$\frac{5 \cdot (1 - \sqrt{x}) \sqrt{x}}{(x - 5) x^{\frac{2}{2}}}$

Step 6.1.7.4.2

Rewrite the expression.

$\frac{5 \cdot (1 - \sqrt{x}) \sqrt{x}}{(x - 5) x^{1}}$

Step 6.1.7.5

Simplify.

$\frac{5 \cdot (1 - \sqrt{x}) \sqrt{x}}{(x - 5) x}$

Step 6.2

Simplify by multiplying through.

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Step 6.2.1

Group $\sqrt{x}$ and $1 - \sqrt{x}$ together.

$\frac{5 \cdot (\sqrt{x} (1 - \sqrt{x}))}{(x - 5) x}$

Step 6.2.2

Apply the distributive property.

$\frac{5 \cdot (\sqrt{x} \cdot 1 + \sqrt{x} (- \sqrt{x}))}{(x - 5) x}$

Step 6.2.3

Simplify the expression.

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Step 6.2.3.1

Multiply $\sqrt{x}$ by $1$ .

$\frac{5 \cdot (\sqrt{x} + \sqrt{x} (- \sqrt{x}))}{(x - 5) x}$

Step 6.2.3.2

Rewrite using the commutative property of multiplication.

$\frac{5 \cdot (\sqrt{x} - \sqrt{x} \sqrt{x})}{(x - 5) x}$

Step 6.3

Simplify each term.

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Step 6.3.1

Multiply $- \sqrt{x} \sqrt{x}$ .

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Step 6.3.1.1

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{5 \cdot (\sqrt{x} - ({\sqrt{x}}^{1} \sqrt{x}))}{(x - 5) x}$

Step 6.3.1.2

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{5 \cdot (\sqrt{x} - ({\sqrt{x}}^{1} {\sqrt{x}}^{1}))}{(x - 5) x}$

Step 6.3.1.3

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

$\frac{5 \cdot (\sqrt{x} - {\sqrt{x}}^{1 + 1})}{(x - 5) x}$

Step 6.3.1.4

Add $1$ and $1$ .

$\frac{5 \cdot (\sqrt{x} - {\sqrt{x}}^{2})}{(x - 5) x}$

Step 6.3.2

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 6.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{5 \cdot (\sqrt{x} - {(x^{\frac{1}{2}})}^{2})}{(x - 5) x}$

Step 6.3.2.2

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

$\frac{5 \cdot (\sqrt{x} - x^{\frac{1}{2} \cdot 2})}{(x - 5) x}$

Step 6.3.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{5 \cdot (\sqrt{x} - x^{\frac{2}{2}})}{(x - 5) x}$

Step 6.3.2.4

Cancel the common factor of $2$ .

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Step 6.3.2.4.1

Cancel the common factor.

$\frac{5 \cdot (\sqrt{x} - x^{\frac{2}{2}})}{(x - 5) x}$

Step 6.3.2.4.2

Rewrite the expression.

$\frac{5 \cdot (\sqrt{x} - x^{1})}{(x - 5) x}$

Step 6.3.2.5

Simplify.

$\frac{5 \cdot (\sqrt{x} - x)}{(x - 5) x}$

Step 7

Simplify the numerator.

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Step 7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{5 (x^{\frac{1}{2}} - x)}{(x - 5) x}$

Step 7.2

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} - x$ .

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Step 7.2.1

Multiply by $1$ .

$\frac{5 (x^{\frac{1}{2}} \cdot 1 - x)}{(x - 5) x}$

Step 7.2.2

Factor $x^{\frac{1}{2}}$ out of $- x$ .

$\frac{5 (x^{\frac{1}{2}} \cdot 1 + x^{\frac{1}{2}} (- x^{\frac{1}{2}}))}{(x - 5) x}$

Step 7.2.3

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} \cdot 1 + x^{\frac{1}{2}} (- x^{\frac{1}{2}})$ .

$\frac{5 (x^{\frac{1}{2}} (1 - x^{\frac{1}{2}}))}{(x - 5) x}$

$\frac{5 x^{\frac{1}{2}} (1 - x^{\frac{1}{2}})}{(x - 5) x}$

Step 8

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{5 (1 - x^{\frac{1}{2}})}{(x - 5) x \cdot x^{- \frac{1}{2}}}$

Step 9

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 9.1

Move $x^{- \frac{1}{2}}$ .

$\frac{5 (1 - x^{\frac{1}{2}})}{(x - 5) (x^{- \frac{1}{2}} x)}$

Step 9.2

Multiply $x^{- \frac{1}{2}}$ by $x$ .

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Step 9.2.1

Raise $x$ to the power of $1$ .

$\frac{5 (1 - x^{\frac{1}{2}})}{(x - 5) (x^{- \frac{1}{2}} x^{1})}$

Step 9.2.2

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

$\frac{5 (1 - x^{\frac{1}{2}})}{(x - 5) x^{- \frac{1}{2} + 1}}$

Step 9.3

Write $1$ as a fraction with a common denominator.

$\frac{5 (1 - x^{\frac{1}{2}})}{(x - 5) x^{- \frac{1}{2} + \frac{2}{2}}}$

Step 9.4

Combine the numerators over the common denominator.

$\frac{5 (1 - x^{\frac{1}{2}})}{(x - 5) x^{\frac{- 1 + 2}{2}}}$

Step 9.5

Add $- 1$ and $2$ .

$\frac{5 (1 - x^{\frac{1}{2}})}{(x - 5) x^{\frac{1}{2}}}$

Step 10

Reorder factors in $\frac{5 (1 - x^{\frac{1}{2}})}{(x - 5) x^{\frac{1}{2}}}$ .

$\frac{5 (1 - x^{\frac{1}{2}})}{x^{\frac{1}{2}} (x - 5)}$