Algebra Examples

$\frac{8 \sqrt{x^{7}} - 6 \sqrt{x^{5}}}{2 \sqrt{x^{3}}}$

Step 1

Cancel the common factor of $8 \sqrt{x^{7}} - 6 \sqrt{x^{5}}$ and $2$ .

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Step 1.1

Factor $2$ out of $8 \sqrt{x^{7}}$ .

$\frac{2 (4 \sqrt{x^{7}}) - 6 \sqrt{x^{5}}}{2 \sqrt{x^{3}}}$

Step 1.2

Factor $2$ out of $- 6 \sqrt{x^{5}}$ .

$\frac{2 (4 \sqrt{x^{7}}) + 2 (- 3 \sqrt{x^{5}})}{2 \sqrt{x^{3}}}$

Step 1.3

Factor $2$ out of $2 (4 \sqrt{x^{7}}) + 2 (- 3 \sqrt{x^{5}})$ .

$\frac{2 (4 \sqrt{x^{7}} - 3 \sqrt{x^{5}})}{2 \sqrt{x^{3}}}$

Step 1.4

Cancel the common factors.

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Step 1.4.1

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

$\frac{2 (4 \sqrt{x^{7}} - 3 \sqrt{x^{5}})}{2 (\sqrt{x^{3}})}$

Step 1.4.2

Cancel the common factor.

$\frac{2 (4 \sqrt{x^{7}} - 3 \sqrt{x^{5}})}{2 \sqrt{x^{3}}}$

Step 1.4.3

Rewrite the expression.

$\frac{4 \sqrt{x^{7}} - 3 \sqrt{x^{5}}}{\sqrt{x^{3}}}$

Step 2

Simplify the numerator.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{7}}$ as $x^{\frac{7}{2}}$ .

$\frac{4 x^{\frac{7}{2}} - 3 \sqrt{x^{5}}}{\sqrt{x^{3}}}$

Step 2.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{5}}$ as $x^{\frac{5}{2}}$ .

$\frac{4 x^{\frac{7}{2}} - 3 x^{\frac{5}{2}}}{\sqrt{x^{3}}}$

Step 2.3

Factor $x^{\frac{5}{2}}$ out of $4 x^{\frac{7}{2}} - 3 x^{\frac{5}{2}}$ .

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Step 2.3.1

Factor $x^{\frac{5}{2}}$ out of $4 x^{\frac{7}{2}}$ .

$\frac{x^{\frac{5}{2}} (4 x^{\frac{2}{2}}) - 3 x^{\frac{5}{2}}}{\sqrt{x^{3}}}$

Step 2.3.2

Factor $x^{\frac{5}{2}}$ out of $- 3 x^{\frac{5}{2}}$ .

$\frac{x^{\frac{5}{2}} (4 x^{\frac{2}{2}}) + x^{\frac{5}{2}} \cdot - 3}{\sqrt{x^{3}}}$

Step 2.3.3

Factor $x^{\frac{5}{2}}$ out of $x^{\frac{5}{2}} (4 x^{\frac{2}{2}}) + x^{\frac{5}{2}} \cdot - 3$ .

$\frac{x^{\frac{5}{2}} (4 x^{\frac{2}{2}} - 3)}{\sqrt{x^{3}}}$

Step 2.4

Divide $2$ by $2$ .

$\frac{x^{\frac{5}{2}} (4 x^{1} - 3)}{\sqrt{x^{3}}}$

Step 2.5

Simplify.

$\frac{x^{\frac{5}{2}} (4 x - 3)}{\sqrt{x^{3}}}$

Step 3

Simplify the denominator.

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Step 3.1

Factor out $x^{2}$ .

$\frac{x^{\frac{5}{2}} (4 x - 3)}{\sqrt{x^{2} x}}$

Step 3.2

Pull terms out from under the radical.

$\frac{x^{\frac{5}{2}} (4 x - 3)}{x \sqrt{x}}$

Step 4

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

$\frac{x^{\frac{5}{2}} (4 x - 3) x^{- 1}}{\sqrt{x}}$

Step 5

Multiply $x^{\frac{5}{2}}$ by $x^{- 1}$ by adding the exponents.

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Step 5.1

Move $x^{- 1}$ .

$\frac{x^{- 1} x^{\frac{5}{2}} (4 x - 3)}{\sqrt{x}}$

Step 5.2

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

$\frac{x^{- 1 + \frac{5}{2}} (4 x - 3)}{\sqrt{x}}$

Step 5.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x^{- 1 \cdot \frac{2}{2} + \frac{5}{2}} (4 x - 3)}{\sqrt{x}}$

Step 5.4

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

$\frac{x^{\frac{- 1 \cdot 2}{2} + \frac{5}{2}} (4 x - 3)}{\sqrt{x}}$

Step 5.5

Combine the numerators over the common denominator.

$\frac{x^{\frac{- 1 \cdot 2 + 5}{2}} (4 x - 3)}{\sqrt{x}}$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Multiply $- 1$ by $2$ .

$\frac{x^{\frac{- 2 + 5}{2}} (4 x - 3)}{\sqrt{x}}$

Step 5.6.2

Add $- 2$ and $5$ .

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

Step 6

Multiply $\frac{x^{\frac{3}{2}} (4 x - 3)}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{x^{\frac{3}{2}} (4 x - 3)}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{x^{\frac{3}{2}} (4 x - 3)}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

Add $1$ and $1$ .

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

Step 7.6

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

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Step 7.6.1

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

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

Step 7.6.2

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

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

Step 7.6.3

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

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

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

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

Step 7.6.4.2

Rewrite the expression.

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

Step 7.6.5

Simplify.

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

Step 8

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

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

Step 9

Multiply $x^{\frac{3}{2}}$ by $x^{- 1}$ by adding the exponents.

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Step 9.1

Move $x^{- 1}$ .

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

Step 9.2

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

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

Step 9.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 9.4

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

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

Step 9.5

Combine the numerators over the common denominator.

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

Step 9.6

Simplify the numerator.

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Step 9.6.1

Multiply $- 1$ by $2$ .

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

Step 9.6.2

Add $- 2$ and $3$ .

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

Step 10

Simplify by multiplying through.

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Step 10.1

Apply the distributive property.

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

Step 10.2

Reorder.

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Step 10.2.1

Rewrite using the commutative property of multiplication.

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

Step 10.2.2

Move $- 3$ to the left of $x^{\frac{1}{2}}$ .

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

Step 11

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

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Step 11.1

Move $x$ .

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

Step 11.2

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

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Step 11.2.1

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

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

Step 11.2.2

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

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

Step 11.3

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

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

Step 11.4

Combine the numerators over the common denominator.

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

Step 11.5

Add $2$ and $1$ .

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

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

Step 12

Apply the distributive property.

$4 x^{\frac{3}{2}} \sqrt{x} - 3 x^{\frac{1}{2}} \sqrt{x}$

Step 13

Multiply $4 x^{\frac{3}{2}} \sqrt{x}$ .

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Step 13.1

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

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

Step 13.2

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

$4 x^{\frac{1}{2} + \frac{3}{2}} - 3 x^{\frac{1}{2}} \sqrt{x}$

Step 13.3

Combine the numerators over the common denominator.

$4 x^{\frac{1 + 3}{2}} - 3 x^{\frac{1}{2}} \sqrt{x}$

Step 13.4

Add $1$ and $3$ .

$4 x^{\frac{4}{2}} - 3 x^{\frac{1}{2}} \sqrt{x}$

Step 13.5

Cancel the common factor of $4$ and $2$ .

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Step 13.5.1

Factor $2$ out of $4$ .

$4 x^{\frac{2 \cdot 2}{2}} - 3 x^{\frac{1}{2}} \sqrt{x}$

Step 13.5.2

Cancel the common factors.

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Step 13.5.2.1

Factor $2$ out of $2$ .

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

Step 13.5.2.2

Cancel the common factor.

$4 x^{\frac{2 \cdot 2}{2 \cdot 1}} - 3 x^{\frac{1}{2}} \sqrt{x}$

Step 13.5.2.3

Rewrite the expression.

$4 x^{\frac{2}{1}} - 3 x^{\frac{1}{2}} \sqrt{x}$

Step 13.5.2.4

Divide $2$ by $1$ .

$4 x^{2} - 3 x^{\frac{1}{2}} \sqrt{x}$

Step 14

Multiply $- 3 x^{\frac{1}{2}} \sqrt{x}$ .

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Step 14.1

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

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

Step 14.2

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

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

Step 14.3

Combine the numerators over the common denominator.

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

Step 14.4

Add $1$ and $1$ .

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

Step 14.5

Cancel the common factor of $2$ .

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Step 14.5.1

Cancel the common factor.

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

Step 14.5.2

Rewrite the expression.

$4 x^{2} - 3 x^{1}$

Step 15

Simplify.

$4 x^{2} - 3 x$