Algebra Examples

$\frac{3 \sqrt{m^{5}} + 5 \sqrt{m^{3}}}{\sqrt{m}}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{3 m^{\frac{5}{2}} + 5 \sqrt{m^{3}}}{\sqrt{m}}$

Step 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{m^{3}}$ as $m^{\frac{3}{2}}$ .

$\frac{3 m^{\frac{5}{2}} + 5 m^{\frac{3}{2}}}{\sqrt{m}}$

Step 1.3

Factor $m^{\frac{3}{2}}$ out of $3 m^{\frac{5}{2}} + 5 m^{\frac{3}{2}}$ .

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Step 1.3.1

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

$\frac{m^{\frac{3}{2}} (3 m^{\frac{2}{2}}) + 5 m^{\frac{3}{2}}}{\sqrt{m}}$

Step 1.3.2

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

$\frac{m^{\frac{3}{2}} (3 m^{\frac{2}{2}}) + m^{\frac{3}{2}} \cdot 5}{\sqrt{m}}$

Step 1.3.3

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

$\frac{m^{\frac{3}{2}} (3 m^{\frac{2}{2}} + 5)}{\sqrt{m}}$

Step 1.4

Divide $2$ by $2$ .

$\frac{m^{\frac{3}{2}} (3 m^{1} + 5)}{\sqrt{m}}$

Step 1.5

Simplify.

$\frac{m^{\frac{3}{2}} (3 m + 5)}{\sqrt{m}}$

Step 2

Multiply $\frac{m^{\frac{3}{2}} (3 m + 5)}{\sqrt{m}}$ by $\frac{\sqrt{m}}{\sqrt{m}}$ .

$\frac{m^{\frac{3}{2}} (3 m + 5)}{\sqrt{m}} \cdot \frac{\sqrt{m}}{\sqrt{m}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{m^{\frac{3}{2}} (3 m + 5)}{\sqrt{m}}$ by $\frac{\sqrt{m}}{\sqrt{m}}$ .

$\frac{m^{\frac{3}{2}} (3 m + 5) \sqrt{m}}{\sqrt{m} \sqrt{m}}$

Step 3.2

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

$\frac{m^{\frac{3}{2}} (3 m + 5) \sqrt{m}}{{\sqrt{m}}^{1} \sqrt{m}}$

Step 3.3

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

$\frac{m^{\frac{3}{2}} (3 m + 5) \sqrt{m}}{{\sqrt{m}}^{1} {\sqrt{m}}^{1}}$

Step 3.4

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

$\frac{m^{\frac{3}{2}} (3 m + 5) \sqrt{m}}{{\sqrt{m}}^{1 + 1}}$

Step 3.5

Add $1$ and $1$ .

$\frac{m^{\frac{3}{2}} (3 m + 5) \sqrt{m}}{{\sqrt{m}}^{2}}$

Step 3.6

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

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Step 3.6.1

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

$\frac{m^{\frac{3}{2}} (3 m + 5) \sqrt{m}}{{(m^{\frac{1}{2}})}^{2}}$

Step 3.6.2

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

$\frac{m^{\frac{3}{2}} (3 m + 5) \sqrt{m}}{m^{\frac{1}{2} \cdot 2}}$

Step 3.6.3

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

$\frac{m^{\frac{3}{2}} (3 m + 5) \sqrt{m}}{m^{\frac{2}{2}}}$

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

$\frac{m^{\frac{3}{2}} (3 m + 5) \sqrt{m}}{m^{\frac{2}{2}}}$

Step 3.6.4.2

Rewrite the expression.

$\frac{m^{\frac{3}{2}} (3 m + 5) \sqrt{m}}{m^{1}}$

Step 3.6.5

Simplify.

$\frac{m^{\frac{3}{2}} (3 m + 5) \sqrt{m}}{m}$

Step 4

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

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

Step 5

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

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Step 5.1

Move $m^{- 1}$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Combine the numerators over the common denominator.

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

Step 5.6

Simplify the numerator.

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Step 5.6.1

Multiply $- 1$ by $2$ .

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

Step 5.6.2

Add $- 2$ and $3$ .

$m^{\frac{1}{2}} (3 m + 5) \sqrt{m}$

Step 6

Simplify by multiplying through.

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Step 6.1

Apply the distributive property.

$(m^{\frac{1}{2}} (3 m) + m^{\frac{1}{2}} \cdot 5) \sqrt{m}$

Step 6.2

Reorder.

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Step 6.2.1

Rewrite using the commutative property of multiplication.

$(3 m^{\frac{1}{2}} m + m^{\frac{1}{2}} \cdot 5) \sqrt{m}$

Step 6.2.2

Move $5$ to the left of $m^{\frac{1}{2}}$ .

$(3 m^{\frac{1}{2}} m + 5 \cdot m^{\frac{1}{2}}) \sqrt{m}$

Step 7

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

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Step 7.1

Move $m$ .

$(3 (m \cdot m^{\frac{1}{2}}) + 5 \cdot m^{\frac{1}{2}}) \sqrt{m}$

Step 7.2

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

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Step 7.2.1

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

$(3 (m^{1} m^{\frac{1}{2}}) + 5 \cdot m^{\frac{1}{2}}) \sqrt{m}$

Step 7.2.2

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

$(3 m^{1 + \frac{1}{2}} + 5 \cdot m^{\frac{1}{2}}) \sqrt{m}$

Step 7.3

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

$(3 m^{\frac{2}{2} + \frac{1}{2}} + 5 \cdot m^{\frac{1}{2}}) \sqrt{m}$

Step 7.4

Combine the numerators over the common denominator.

$(3 m^{\frac{2 + 1}{2}} + 5 \cdot m^{\frac{1}{2}}) \sqrt{m}$

Step 7.5

Add $2$ and $1$ .

$(3 m^{\frac{3}{2}} + 5 \cdot m^{\frac{1}{2}}) \sqrt{m}$

$(3 m^{\frac{3}{2}} + 5 m^{\frac{1}{2}}) \sqrt{m}$

Step 8

Apply the distributive property.

$3 m^{\frac{3}{2}} \sqrt{m} + 5 m^{\frac{1}{2}} \sqrt{m}$

Step 9

Multiply $3 m^{\frac{3}{2}} \sqrt{m}$ .

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Step 9.1

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

$3 (m^{\frac{1}{2}} m^{\frac{3}{2}}) + 5 m^{\frac{1}{2}} \sqrt{m}$

Step 9.2

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

$3 m^{\frac{1}{2} + \frac{3}{2}} + 5 m^{\frac{1}{2}} \sqrt{m}$

Step 9.3

Combine the numerators over the common denominator.

$3 m^{\frac{1 + 3}{2}} + 5 m^{\frac{1}{2}} \sqrt{m}$

Step 9.4

Add $1$ and $3$ .

$3 m^{\frac{4}{2}} + 5 m^{\frac{1}{2}} \sqrt{m}$

Step 9.5

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

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Step 9.5.1

Factor $2$ out of $4$ .

$3 m^{\frac{2 \cdot 2}{2}} + 5 m^{\frac{1}{2}} \sqrt{m}$

Step 9.5.2

Cancel the common factors.

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Step 9.5.2.1

Factor $2$ out of $2$ .

$3 m^{\frac{2 \cdot 2}{2 (1)}} + 5 m^{\frac{1}{2}} \sqrt{m}$

Step 9.5.2.2

Cancel the common factor.

$3 m^{\frac{2 \cdot 2}{2 \cdot 1}} + 5 m^{\frac{1}{2}} \sqrt{m}$

Step 9.5.2.3

Rewrite the expression.

$3 m^{\frac{2}{1}} + 5 m^{\frac{1}{2}} \sqrt{m}$

Step 9.5.2.4

Divide $2$ by $1$ .

$3 m^{2} + 5 m^{\frac{1}{2}} \sqrt{m}$

Step 10

Multiply $5 m^{\frac{1}{2}} \sqrt{m}$ .

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Step 10.1

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

$3 m^{2} + 5 (m^{\frac{1}{2}} m^{\frac{1}{2}})$

Step 10.2

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

$3 m^{2} + 5 m^{\frac{1}{2} + \frac{1}{2}}$

Step 10.3

Combine the numerators over the common denominator.

$3 m^{2} + 5 m^{\frac{1 + 1}{2}}$

Step 10.4

Add $1$ and $1$ .

$3 m^{2} + 5 m^{\frac{2}{2}}$

Step 10.5

Cancel the common factor of $2$ .

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Step 10.5.1

Cancel the common factor.

$3 m^{2} + 5 m^{\frac{2}{2}}$

Step 10.5.2

Rewrite the expression.

$3 m^{2} + 5 m^{1}$

Step 11

Simplify.

$3 m^{2} + 5 m$