Algebra Examples

$\sqrt{3 \frac{4}{9} \cdot 7 \frac{8}{31}}$

Step 1

Convert $3 \frac{4}{9}$ to an improper fraction.

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Step 1.1

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

$\sqrt{(3 + \frac{4}{9}) \cdot 7 \frac{8}{31}}$

Step 1.2

Add $3$ and $\frac{4}{9}$ .

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Step 1.2.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\sqrt{(3 \cdot \frac{9}{9} + \frac{4}{9}) \cdot 7 \frac{8}{31}}$

Step 1.2.2

Combine $3$ and $\frac{9}{9}$ .

$\sqrt{(\frac{3 \cdot 9}{9} + \frac{4}{9}) \cdot 7 \frac{8}{31}}$

Step 1.2.3

Combine the numerators over the common denominator.

$\sqrt{\frac{3 \cdot 9 + 4}{9} \cdot 7 \frac{8}{31}}$

Step 1.2.4

Simplify the numerator.

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Step 1.2.4.1

Multiply $3$ by $9$ .

$\sqrt{\frac{27 + 4}{9} \cdot 7 \frac{8}{31}}$

Step 1.2.4.2

Add $27$ and $4$ .

$\sqrt{\frac{31}{9} \cdot 7 \frac{8}{31}}$

Step 2

Convert $7 \frac{8}{31}$ to an improper fraction.

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Step 2.1

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

$\sqrt{\frac{31}{9} \cdot (7 + \frac{8}{31})}$

Step 2.2

Add $7$ and $\frac{8}{31}$ .

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Step 2.2.1

To write $7$ as a fraction with a common denominator, multiply by $\frac{31}{31}$ .

$\sqrt{\frac{31}{9} \cdot (7 \cdot \frac{31}{31} + \frac{8}{31})}$

Step 2.2.2

Combine $7$ and $\frac{31}{31}$ .

$\sqrt{\frac{31}{9} \cdot (\frac{7 \cdot 31}{31} + \frac{8}{31})}$

Step 2.2.3

Combine the numerators over the common denominator.

$\sqrt{\frac{31}{9} \cdot \frac{7 \cdot 31 + 8}{31}}$

Step 2.2.4

Simplify the numerator.

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Step 2.2.4.1

Multiply $7$ by $31$ .

$\sqrt{\frac{31}{9} \cdot \frac{217 + 8}{31}}$

Step 2.2.4.2

Add $217$ and $8$ .

$\sqrt{\frac{31}{9} \cdot \frac{225}{31}}$

Step 3

Multiply $\frac{31}{9}$ by $\frac{225}{31}$ .

$\sqrt{\frac{31 \cdot 225}{9 \cdot 31}}$

Step 4

Reduce the expression $\frac{31 \cdot 225}{9 \cdot 31}$ by cancelling the common factors.

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Step 4.1

Cancel the common factor.

$\sqrt{\frac{31 \cdot 225}{9 \cdot 31}}$

Step 4.2

Rewrite the expression.

$\sqrt{\frac{225}{9}}$

Step 5

Simplify the expression.

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Step 5.1

Divide $225$ by $9$ .

$\sqrt{25}$

Step 5.2

Rewrite $25$ as $5^{2}$ .

$\sqrt{5^{2}}$

Step 6

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

$5$