Simplify (3 square root of m^5+5 square root of m^3)/( square root of m)

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

Step 1

Simplify the numerator.

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

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

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

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

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

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

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

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

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

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

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

Divide $2$ by $2$ .

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

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.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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.

Move $m^{- 1}$ .

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Add $- 2$ and $3$ .

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

Step 6

Simplify by multiplying through.

Apply the distributive property.

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

Reorder.

Rewrite using the commutative property of multiplication.

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

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.

Move $m$ .

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $3$ .

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

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

Factor $2$ out of $4$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $1$ .

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

Step 10

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 11

Simplify.

$3 m^{2} + 5 m$

Simplify (3 square root of m^5+5 square root of m^3)/( square root of m)

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