Basic Math Examples

$\frac{\sqrt{72}}{3 y} + \frac{3 \sqrt{8}}{4 y}$

Step 1

Simplify each term.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 1.1.1.1

Factor $36$ out of $72$ .

$\frac{\sqrt{36 (2)}}{3 y} + \frac{3 \sqrt{8}}{4 y}$

Step 1.1.1.2

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{6^{2} \cdot 2}}{3 y} + \frac{3 \sqrt{8}}{4 y}$

Step 1.1.2

Pull terms out from under the radical.

$\frac{6 \sqrt{2}}{3 y} + \frac{3 \sqrt{8}}{4 y}$

Step 1.2

Cancel the common factor of $6$ and $3$ .

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Step 1.2.1

Factor $3$ out of $6 \sqrt{2}$ .

$\frac{3 (2 \sqrt{2})}{3 y} + \frac{3 \sqrt{8}}{4 y}$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $3$ out of $3 y$ .

$\frac{3 (2 \sqrt{2})}{3 (y)} + \frac{3 \sqrt{8}}{4 y}$

Step 1.2.2.2

Cancel the common factor.

$\frac{3 (2 \sqrt{2})}{3 y} + \frac{3 \sqrt{8}}{4 y}$

Step 1.2.2.3

Rewrite the expression.

$\frac{2 \sqrt{2}}{y} + \frac{3 \sqrt{8}}{4 y}$

Step 1.3

Simplify the numerator.

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Step 1.3.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.3.1.1

Factor $4$ out of $8$ .

$\frac{2 \sqrt{2}}{y} + \frac{3 \sqrt{4 (2)}}{4 y}$

Step 1.3.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{2 \sqrt{2}}{y} + \frac{3 \sqrt{2^{2} \cdot 2}}{4 y}$

Step 1.3.2

Pull terms out from under the radical.

$\frac{2 \sqrt{2}}{y} + \frac{3 \cdot 2 \sqrt{2}}{4 y}$

Step 1.3.3

Multiply $3$ by $2$ .

$\frac{2 \sqrt{2}}{y} + \frac{6 \sqrt{2}}{4 y}$

Step 1.4

Cancel the common factor of $6$ and $4$ .

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Step 1.4.1

Factor $2$ out of $6 \sqrt{2}$ .

$\frac{2 \sqrt{2}}{y} + \frac{2 (3 \sqrt{2})}{4 y}$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Factor $2$ out of $4 y$ .

$\frac{2 \sqrt{2}}{y} + \frac{2 (3 \sqrt{2})}{2 (2 y)}$

Step 1.4.2.2

Cancel the common factor.

$\frac{2 \sqrt{2}}{y} + \frac{2 (3 \sqrt{2})}{2 (2 y)}$

Step 1.4.2.3

Rewrite the expression.

$\frac{2 \sqrt{2}}{y} + \frac{3 \sqrt{2}}{2 y}$

Step 2

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

$\frac{2 \sqrt{2}}{y} \cdot \frac{2}{2} + \frac{3 \sqrt{2}}{2 y}$

Step 3

Write each expression with a common denominator of $y \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{2 \sqrt{2}}{y}$ by $\frac{2}{2}$ .

$\frac{2 \sqrt{2} \cdot 2}{y \cdot 2} + \frac{3 \sqrt{2}}{2 y}$

Step 3.2

Move $2$ to the left of $y$ .

$\frac{2 \sqrt{2} \cdot 2}{2 y} + \frac{3 \sqrt{2}}{2 y}$

Step 4

Combine the numerators over the common denominator.

$\frac{2 \sqrt{2} \cdot 2 + 3 \sqrt{2}}{2 y}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $2$ by $2$ .

$\frac{4 \sqrt{2} + 3 \sqrt{2}}{2 y}$

Step 5.2

Add $4 \sqrt{2}$ and $3 \sqrt{2}$ .

$\frac{7 \sqrt{2}}{2 y}$