Algebra Examples

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

Step 1

To write $\sqrt{2 a}$ as a fraction with a common denominator, multiply by $\frac{5 m}{5 m}$ .

$\frac{3 a b^{2}}{5 m} + \sqrt{2 a} \cdot \frac{5 m}{5 m} - a^{2} m^{2}$

Step 2

Simplify terms.

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Step 2.1

Combine $\sqrt{2 a}$ and $\frac{5 m}{5 m}$ .

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

Apply the product rule to $2 a$ .

$\frac{3 a b^{2} + 2^{\frac{1}{2}} a^{\frac{1}{2}} \cdot 5 m}{5 m} - a^{2} m^{2}$

Step 3.3

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

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

Step 3.4

Factor $a^{\frac{1}{2}}$ out of $3 a b^{2} + 5 \cdot 2^{\frac{1}{2}} a^{\frac{1}{2}} m$ .

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Step 3.4.1

Factor $a^{\frac{1}{2}}$ out of $3 a b^{2}$ .

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

Step 3.4.2

Factor $a^{\frac{1}{2}}$ out of $5 \cdot 2^{\frac{1}{2}} a^{\frac{1}{2}} m$ .

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

Step 3.4.3

Factor $a^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} (3 a^{\frac{1}{2}} b^{2}) + a^{\frac{1}{2}} (5 \cdot 2^{\frac{1}{2}} m)$ .

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

Step 4

To write $- a^{2} m^{2}$ as a fraction with a common denominator, multiply by $\frac{5 m}{5 m}$ .

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

Step 5

Simplify terms.

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Step 5.1

Combine $- a^{2} m^{2}$ and $\frac{5 m}{5 m}$ .

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Factor $a^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} (3 a^{\frac{1}{2}} b^{2} + 5 \cdot 2^{\frac{1}{2}} m) - a^{2} m^{2} (5 m)$ .

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Step 6.1.1

Reorder the expression.

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Step 6.1.1.1

Move $m^{2}$ .

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

Step 6.1.1.2

Move $a^{2}$ .

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

Step 6.1.2

Factor $a^{\frac{1}{2}}$ out of $- 1 \cdot 5 a^{2} m^{2} m$ .

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

Step 6.1.3

Factor $a^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} (3 a^{\frac{1}{2}} b^{2} + 5 \cdot 2^{\frac{1}{2}} m) + a^{\frac{1}{2}} (- 1 \cdot 5 a^{\frac{3}{2}} m^{2} m)$ .

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

Step 6.2

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

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Step 6.2.1

Move $m$ .

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

Step 6.2.2

Multiply $m$ by $m^{2}$ .

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Step 6.2.2.1

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

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

Step 6.2.2.2

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

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

Step 6.2.3

Add $1$ and $2$ .

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

Step 6.3

Multiply $- 1$ by $5$ .

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

Step 7

Reorder factors in $\frac{a^{\frac{1}{2}} (3 a^{\frac{1}{2}} b^{2} + 5 \cdot 2^{\frac{1}{2}} m - 5 a^{\frac{3}{2}} m^{3})}{5 m}$ .

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