Algebra Examples

$\frac{5}{7} \sqrt{63 n^{3}} - 2 \sqrt{112 n^{3}}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $63 n^{3}$ as ${(3 n)}^{2} \cdot (7 n)$ .

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Step 1.1.1

Factor $9$ out of $63$ .

$\frac{5}{7} \cdot \sqrt{9 (7) n^{3}} - 2 \sqrt{112 n^{3}}$

Step 1.1.2

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

$\frac{5}{7} \cdot \sqrt{3^{2} \cdot 7 n^{3}} - 2 \sqrt{112 n^{3}}$

Step 1.1.3

Factor out $n^{2}$ .

$\frac{5}{7} \cdot \sqrt{3^{2} \cdot 7 (n^{2} n)} - 2 \sqrt{112 n^{3}}$

Step 1.1.4

Move $7$ .

$\frac{5}{7} \cdot \sqrt{3^{2} n^{2} \cdot 7 n} - 2 \sqrt{112 n^{3}}$

Step 1.1.5

Rewrite $3^{2} n^{2}$ as ${(3 n)}^{2}$ .

$\frac{5}{7} \cdot \sqrt{{(3 n)}^{2} \cdot 7 n} - 2 \sqrt{112 n^{3}}$

Step 1.1.6

Add parentheses.

$\frac{5}{7} \cdot \sqrt{{(3 n)}^{2} \cdot (7 n)} - 2 \sqrt{112 n^{3}}$

Step 1.2

Pull terms out from under the radical.

$\frac{5}{7} \cdot (3 n \sqrt{7 n}) - 2 \sqrt{112 n^{3}}$

Step 1.3

Multiply $\frac{5}{7} (3 n \sqrt{7 n})$ .

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Step 1.3.1

Combine $3$ and $\frac{5}{7}$ .

$\frac{3 \cdot 5}{7} (n \sqrt{7 n}) - 2 \sqrt{112 n^{3}}$

Step 1.3.2

Multiply $3$ by $5$ .

$\frac{15}{7} (n \sqrt{7 n}) - 2 \sqrt{112 n^{3}}$

Step 1.3.3

Combine $n$ and $\frac{15}{7}$ .

$\frac{n \cdot 15}{7} \sqrt{7 n} - 2 \sqrt{112 n^{3}}$

Step 1.3.4

Combine $\frac{n \cdot 15}{7}$ and $\sqrt{7 n}$ .

$\frac{n \cdot 15 \sqrt{7 n}}{7} - 2 \sqrt{112 n^{3}}$

Step 1.4

Move $15$ to the left of $n$ .

$\frac{15 \cdot n \sqrt{7 n}}{7} - 2 \sqrt{112 n^{3}}$

Step 1.5

Rewrite $112 n^{3}$ as ${(4 n)}^{2} \cdot (7 n)$ .

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Step 1.5.1

Factor $16$ out of $112$ .

$\frac{15 n \sqrt{7 n}}{7} - 2 \sqrt{16 (7) n^{3}}$

Step 1.5.2

Rewrite $16$ as $4^{2}$ .

$\frac{15 n \sqrt{7 n}}{7} - 2 \sqrt{4^{2} \cdot 7 n^{3}}$

Step 1.5.3

Factor out $n^{2}$ .

$\frac{15 n \sqrt{7 n}}{7} - 2 \sqrt{4^{2} \cdot 7 (n^{2} n)}$

Step 1.5.4

Move $7$ .

$\frac{15 n \sqrt{7 n}}{7} - 2 \sqrt{4^{2} n^{2} \cdot 7 n}$

Step 1.5.5

Rewrite $4^{2} n^{2}$ as ${(4 n)}^{2}$ .

$\frac{15 n \sqrt{7 n}}{7} - 2 \sqrt{{(4 n)}^{2} \cdot 7 n}$

Step 1.5.6

Add parentheses.

$\frac{15 n \sqrt{7 n}}{7} - 2 \sqrt{{(4 n)}^{2} \cdot (7 n)}$

Step 1.6

Pull terms out from under the radical.

$\frac{15 n \sqrt{7 n}}{7} - 2 (4 n \sqrt{7 n})$

Step 1.7

Multiply $4$ by $- 2$ .

$\frac{15 n \sqrt{7 n}}{7} - 8 n \sqrt{7 n}$

Step 2

To write $- 8 n \sqrt{7 n}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{15 n \sqrt{7 n}}{7} - 8 n \sqrt{7 n} \cdot \frac{7}{7}$

Step 3

Simplify terms.

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Step 3.1

Combine $- 8 n \sqrt{7 n}$ and $\frac{7}{7}$ .

$\frac{15 n \sqrt{7 n}}{7} + \frac{- 8 n \sqrt{7 n} \cdot 7}{7}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{15 n \sqrt{7 n} - 8 n \sqrt{7 n} \cdot 7}{7}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $n \sqrt{7 n}$ out of $15 n \sqrt{7 n} - 8 n \sqrt{7 n} \cdot 7$ .

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Step 4.1.1

Factor $n \sqrt{7 n}$ out of $15 n \sqrt{7 n}$ .

$\frac{n \sqrt{7 n} (15) - 8 n \sqrt{7 n} \cdot 7}{7}$

Step 4.1.2

Factor $n \sqrt{7 n}$ out of $- 8 n \sqrt{7 n} \cdot 7$ .

$\frac{n \sqrt{7 n} (15) + n \sqrt{7 n} (- 8 \cdot 7)}{7}$

Step 4.1.3

Factor $n \sqrt{7 n}$ out of $n \sqrt{7 n} (15) + n \sqrt{7 n} (- 8 \cdot 7)$ .

$\frac{n \sqrt{7 n} (15 - 8 \cdot 7)}{7}$

Step 4.2

Multiply $- 8$ by $7$ .

$\frac{n \sqrt{7 n} (15 - 56)}{7}$

Step 4.3

Subtract $56$ from $15$ .

$\frac{n \sqrt{7 n} \cdot - 41}{7}$

Step 5

Simplify the expression.

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Step 5.1

Move $- 41$ to the left of $n \sqrt{7 n}$ .

$\frac{- 41 \cdot (n \sqrt{7 n})}{7}$

Step 5.2

Move the negative in front of the fraction.

$- \frac{41 n \sqrt{7 n}}{7}$