Basic Math Examples

$\sqrt{\frac{4 \frac{1}{2}}{2}} + \sqrt{\frac{12 \frac{1}{4}}{9}}$

Step 1

Convert $4 \frac{1}{2}$ to an improper fraction.

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Step 1.1

A mixed number is an addition of its whole and fractional parts.

$\sqrt{\frac{4 + \frac{1}{2}}{2}} + \sqrt{\frac{12 \frac{1}{4}}{9}}$

Step 1.2

Add $4$ and $\frac{1}{2}$ .

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Step 1.2.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{\frac{4 \cdot \frac{2}{2} + \frac{1}{2}}{2}} + \sqrt{\frac{12 \frac{1}{4}}{9}}$

Step 1.2.2

Combine $4$ and $\frac{2}{2}$ .

$\sqrt{\frac{\frac{4 \cdot 2}{2} + \frac{1}{2}}{2}} + \sqrt{\frac{12 \frac{1}{4}}{9}}$

Step 1.2.3

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{4 \cdot 2 + 1}{2}}{2}} + \sqrt{\frac{12 \frac{1}{4}}{9}}$

Step 1.2.4

Simplify the numerator.

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Step 1.2.4.1

Multiply $4$ by $2$ .

$\sqrt{\frac{\frac{8 + 1}{2}}{2}} + \sqrt{\frac{12 \frac{1}{4}}{9}}$

Step 1.2.4.2

Add $8$ and $1$ .

$\sqrt{\frac{\frac{9}{2}}{2}} + \sqrt{\frac{12 \frac{1}{4}}{9}}$

Step 2

Convert $12 \frac{1}{4}$ to an improper fraction.

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Step 2.1

A mixed number is an addition of its whole and fractional parts.

$\sqrt{\frac{\frac{9}{2}}{2}} + \sqrt{\frac{12 + \frac{1}{4}}{9}}$

Step 2.2

Add $12$ and $\frac{1}{4}$ .

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Step 2.2.1

To write $12$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{\frac{9}{2}}{2}} + \sqrt{\frac{12 \cdot \frac{4}{4} + \frac{1}{4}}{9}}$

Step 2.2.2

Combine $12$ and $\frac{4}{4}$ .

$\sqrt{\frac{\frac{9}{2}}{2}} + \sqrt{\frac{\frac{12 \cdot 4}{4} + \frac{1}{4}}{9}}$

Step 2.2.3

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{9}{2}}{2}} + \sqrt{\frac{\frac{12 \cdot 4 + 1}{4}}{9}}$

Step 2.2.4

Simplify the numerator.

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Step 2.2.4.1

Multiply $12$ by $4$ .

$\sqrt{\frac{\frac{9}{2}}{2}} + \sqrt{\frac{\frac{48 + 1}{4}}{9}}$

Step 2.2.4.2

Add $48$ and $1$ .

$\sqrt{\frac{\frac{9}{2}}{2}} + \sqrt{\frac{\frac{49}{4}}{9}}$

Step 3

Simplify each term.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{9}{2} \cdot \frac{1}{2}} + \sqrt{\frac{\frac{49}{4}}{9}}$

Step 3.2

Multiply $\frac{9}{2} \cdot \frac{1}{2}$ .

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Step 3.2.1

Multiply $\frac{9}{2}$ by $\frac{1}{2}$ .

$\sqrt{\frac{9}{2 \cdot 2}} + \sqrt{\frac{\frac{49}{4}}{9}}$

Step 3.2.2

Multiply $2$ by $2$ .

$\sqrt{\frac{9}{4}} + \sqrt{\frac{\frac{49}{4}}{9}}$

Step 3.3

Rewrite $\sqrt{\frac{9}{4}}$ as $\frac{\sqrt{9}}{\sqrt{4}}$ .

$\frac{\sqrt{9}}{\sqrt{4}} + \sqrt{\frac{\frac{49}{4}}{9}}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2}}}{\sqrt{4}} + \sqrt{\frac{\frac{49}{4}}{9}}$

Step 3.4.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3}{\sqrt{4}} + \sqrt{\frac{\frac{49}{4}}{9}}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

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

$\frac{3}{\sqrt{2^{2}}} + \sqrt{\frac{\frac{49}{4}}{9}}$

Step 3.5.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3}{2} + \sqrt{\frac{\frac{49}{4}}{9}}$

Step 3.6

Multiply the numerator by the reciprocal of the denominator.

$\frac{3}{2} + \sqrt{\frac{49}{4} \cdot \frac{1}{9}}$

Step 3.7

Multiply $\frac{49}{4} \cdot \frac{1}{9}$ .

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Step 3.7.1

Multiply $\frac{49}{4}$ by $\frac{1}{9}$ .

$\frac{3}{2} + \sqrt{\frac{49}{4 \cdot 9}}$

Step 3.7.2

Multiply $4$ by $9$ .

$\frac{3}{2} + \sqrt{\frac{49}{36}}$

Step 3.8

Rewrite $\sqrt{\frac{49}{36}}$ as $\frac{\sqrt{49}}{\sqrt{36}}$ .

$\frac{3}{2} + \frac{\sqrt{49}}{\sqrt{36}}$

Step 3.9

Simplify the numerator.

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Step 3.9.1

Rewrite $49$ as $7^{2}$ .

$\frac{3}{2} + \frac{\sqrt{7^{2}}}{\sqrt{36}}$

Step 3.9.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3}{2} + \frac{7}{\sqrt{36}}$

Step 3.10

Simplify the denominator.

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Step 3.10.1

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

$\frac{3}{2} + \frac{7}{\sqrt{6^{2}}}$

Step 3.10.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3}{2} + \frac{7}{6}$

Step 4

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{3}{2} \cdot \frac{3}{3} + \frac{7}{6}$

Step 5

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

Multiply $\frac{3}{2}$ by $\frac{3}{3}$ .

$\frac{3 \cdot 3}{2 \cdot 3} + \frac{7}{6}$

Step 5.2

Multiply $2$ by $3$ .

$\frac{3 \cdot 3}{6} + \frac{7}{6}$

Step 6

Combine the numerators over the common denominator.

$\frac{3 \cdot 3 + 7}{6}$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $3$ by $3$ .

$\frac{9 + 7}{6}$

Step 7.2

Add $9$ and $7$ .

$\frac{16}{6}$

Step 8

Cancel the common factor of $16$ and $6$ .

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Step 8.1

Factor $2$ out of $16$ .

$\frac{2 (8)}{6}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 8}{2 \cdot 3}$

Step 8.2.2

Cancel the common factor.

$\frac{2 \cdot 8}{2 \cdot 3}$

Step 8.2.3

Rewrite the expression.

$\frac{8}{3}$