Algebra Examples

$2 \sqrt{32 d^{5}} - \frac{7}{8} \sqrt{8 d^{5}} + d^{2} \sqrt{98 d}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $32 d^{5}$ as ${(4 d^{2})}^{2} \cdot (2 d)$ .

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Step 1.1.1

Factor $16$ out of $32$ .

$2 \sqrt{16 (2) d^{5}} - \frac{7}{8} \cdot \sqrt{8 d^{5}} + d^{2} \sqrt{98 d}$

Step 1.1.2

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

$2 \sqrt{4^{2} \cdot 2 d^{5}} - \frac{7}{8} \cdot \sqrt{8 d^{5}} + d^{2} \sqrt{98 d}$

Step 1.1.3

Factor out $d^{4}$ .

$2 \sqrt{4^{2} \cdot 2 (d^{4} d)} - \frac{7}{8} \cdot \sqrt{8 d^{5}} + d^{2} \sqrt{98 d}$

Step 1.1.4

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

$2 \sqrt{4^{2} \cdot 2 ({(d^{2})}^{2} d)} - \frac{7}{8} \cdot \sqrt{8 d^{5}} + d^{2} \sqrt{98 d}$

Step 1.1.5

Move $2$ .

$2 \sqrt{4^{2} {(d^{2})}^{2} \cdot 2 d} - \frac{7}{8} \cdot \sqrt{8 d^{5}} + d^{2} \sqrt{98 d}$

Step 1.1.6

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

$2 \sqrt{{(4 d^{2})}^{2} \cdot 2 d} - \frac{7}{8} \cdot \sqrt{8 d^{5}} + d^{2} \sqrt{98 d}$

Step 1.1.7

Add parentheses.

$2 \sqrt{{(4 d^{2})}^{2} \cdot (2 d)} - \frac{7}{8} \cdot \sqrt{8 d^{5}} + d^{2} \sqrt{98 d}$

Step 1.2

Pull terms out from under the radical.

$2 (4 d^{2} \sqrt{2 d}) - \frac{7}{8} \cdot \sqrt{8 d^{5}} + d^{2} \sqrt{98 d}$

Step 1.3

Multiply $4$ by $2$ .

$8 (d^{2} \sqrt{2 d}) - \frac{7}{8} \cdot \sqrt{8 d^{5}} + d^{2} \sqrt{98 d}$

Step 1.4

Rewrite $8 d^{5}$ as ${(2 d^{2})}^{2} \cdot (2 d)$ .

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Step 1.4.1

Factor $4$ out of $8$ .

$8 d^{2} \sqrt{2 d} - \frac{7}{8} \cdot \sqrt{4 (2) d^{5}} + d^{2} \sqrt{98 d}$

Step 1.4.2

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

$8 d^{2} \sqrt{2 d} - \frac{7}{8} \cdot \sqrt{2^{2} \cdot 2 d^{5}} + d^{2} \sqrt{98 d}$

Step 1.4.3

Factor out $d^{4}$ .

$8 d^{2} \sqrt{2 d} - \frac{7}{8} \cdot \sqrt{2^{2} \cdot 2 (d^{4} d)} + d^{2} \sqrt{98 d}$

Step 1.4.4

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

$8 d^{2} \sqrt{2 d} - \frac{7}{8} \cdot \sqrt{2^{2} \cdot 2 ({(d^{2})}^{2} d)} + d^{2} \sqrt{98 d}$

Step 1.4.5

Move $2$ .

$8 d^{2} \sqrt{2 d} - \frac{7}{8} \cdot \sqrt{2^{2} {(d^{2})}^{2} \cdot 2 d} + d^{2} \sqrt{98 d}$

Step 1.4.6

Rewrite $2^{2} {(d^{2})}^{2}$ as ${(2 d^{2})}^{2}$ .

$8 d^{2} \sqrt{2 d} - \frac{7}{8} \cdot \sqrt{{(2 d^{2})}^{2} \cdot 2 d} + d^{2} \sqrt{98 d}$

Step 1.4.7

Add parentheses.

$8 d^{2} \sqrt{2 d} - \frac{7}{8} \cdot \sqrt{{(2 d^{2})}^{2} \cdot (2 d)} + d^{2} \sqrt{98 d}$

Step 1.5

Pull terms out from under the radical.

$8 d^{2} \sqrt{2 d} - \frac{7}{8} \cdot (2 d^{2} \sqrt{2 d}) + d^{2} \sqrt{98 d}$

Step 1.6

Cancel the common factor of $2$ .

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Step 1.6.1

Move the leading negative in $- \frac{7}{8}$ into the numerator.

$8 d^{2} \sqrt{2 d} + \frac{- 7}{8} \cdot (2 d^{2} \sqrt{2 d}) + d^{2} \sqrt{98 d}$

Step 1.6.2

Factor $2$ out of $8$ .

$8 d^{2} \sqrt{2 d} + \frac{- 7}{2 (4)} \cdot (2 d^{2} \sqrt{2 d}) + d^{2} \sqrt{98 d}$

Step 1.6.3

Factor $2$ out of $2 d^{2} \sqrt{2 d}$ .

$8 d^{2} \sqrt{2 d} + \frac{- 7}{2 (4)} \cdot (2 (d^{2} \sqrt{2 d})) + d^{2} \sqrt{98 d}$

Step 1.6.4

Cancel the common factor.

$8 d^{2} \sqrt{2 d} + \frac{- 7}{2 \cdot 4} \cdot (2 (d^{2} \sqrt{2 d})) + d^{2} \sqrt{98 d}$

Step 1.6.5

Rewrite the expression.

$8 d^{2} \sqrt{2 d} + \frac{- 7}{4} \cdot (d^{2} \sqrt{2 d}) + d^{2} \sqrt{98 d}$

Step 1.7

Combine $d^{2}$ and $\frac{- 7}{4}$ .

$8 d^{2} \sqrt{2 d} + \frac{d^{2} \cdot - 7}{4} \cdot \sqrt{2 d} + d^{2} \sqrt{98 d}$

Step 1.8

Combine $\frac{d^{2} \cdot - 7}{4}$ and $\sqrt{2 d}$ .

$8 d^{2} \sqrt{2 d} + \frac{d^{2} \cdot - 7 \sqrt{2 d}}{4} + d^{2} \sqrt{98 d}$

Step 1.9

Move $- 7$ to the left of $d^{2}$ .

$8 d^{2} \sqrt{2 d} + \frac{- 7 \cdot d^{2} \sqrt{2 d}}{4} + d^{2} \sqrt{98 d}$

Step 1.10

Move the negative in front of the fraction.

$8 d^{2} \sqrt{2 d} - \frac{7 \cdot d^{2} \sqrt{2 d}}{4} + d^{2} \sqrt{98 d}$

Step 1.11

Rewrite $98 d$ as $7^{2} \cdot (2 d)$ .

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Step 1.11.1

Factor $49$ out of $98$ .

$8 d^{2} \sqrt{2 d} - \frac{7 d^{2} \sqrt{2 d}}{4} + d^{2} \sqrt{49 (2) d}$

Step 1.11.2

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

$8 d^{2} \sqrt{2 d} - \frac{7 d^{2} \sqrt{2 d}}{4} + d^{2} \sqrt{7^{2} \cdot 2 d}$

Step 1.11.3

Add parentheses.

$8 d^{2} \sqrt{2 d} - \frac{7 d^{2} \sqrt{2 d}}{4} + d^{2} \sqrt{7^{2} \cdot (2 d)}$

Step 1.12

Pull terms out from under the radical.

$8 d^{2} \sqrt{2 d} - \frac{7 d^{2} \sqrt{2 d}}{4} + d^{2} (7 \sqrt{2 d})$

Step 1.13

Rewrite using the commutative property of multiplication.

$8 d^{2} \sqrt{2 d} - \frac{7 d^{2} \sqrt{2 d}}{4} + 7 d^{2} \sqrt{2 d}$

Step 2

To write $8 d^{2} \sqrt{2 d}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$8 d^{2} \sqrt{2 d} \cdot \frac{4}{4} - \frac{7 d^{2} \sqrt{2 d}}{4} + 7 d^{2} \sqrt{2 d}$

Step 3

Simplify terms.

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Step 3.1

Combine $8 d^{2} \sqrt{2 d}$ and $\frac{4}{4}$ .

$\frac{8 d^{2} \sqrt{2 d} \cdot 4}{4} - \frac{7 d^{2} \sqrt{2 d}}{4} + 7 d^{2} \sqrt{2 d}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{8 d^{2} \sqrt{2 d} \cdot 4 - 7 d^{2} \sqrt{2 d}}{4} + 7 d^{2} \sqrt{2 d}$

Step 4

Simplify each term.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $d^{2} \sqrt{2 d}$ out of $8 d^{2} \sqrt{2 d} \cdot 4 - 7 d^{2} \sqrt{2 d}$ .

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Step 4.1.1.1

Factor $d^{2} \sqrt{2 d}$ out of $8 d^{2} \sqrt{2 d} \cdot 4$ .

$\frac{d^{2} \sqrt{2 d} (8 \cdot 4) - 7 d^{2} \sqrt{2 d}}{4} + 7 d^{2} \sqrt{2 d}$

Step 4.1.1.2

Factor $d^{2} \sqrt{2 d}$ out of $- 7 d^{2} \sqrt{2 d}$ .

$\frac{d^{2} \sqrt{2 d} (8 \cdot 4) + d^{2} \sqrt{2 d} (- 7)}{4} + 7 d^{2} \sqrt{2 d}$

Step 4.1.1.3

Factor $d^{2} \sqrt{2 d}$ out of $d^{2} \sqrt{2 d} (8 \cdot 4) + d^{2} \sqrt{2 d} (- 7)$ .

$\frac{d^{2} \sqrt{2 d} (8 \cdot 4 - 7)}{4} + 7 d^{2} \sqrt{2 d}$

Step 4.1.2

Multiply $8$ by $4$ .

$\frac{d^{2} \sqrt{2 d} (32 - 7)}{4} + 7 d^{2} \sqrt{2 d}$

Step 4.1.3

Subtract $7$ from $32$ .

$\frac{d^{2} \sqrt{2 d} \cdot 25}{4} + 7 d^{2} \sqrt{2 d}$

Step 4.2

Move $25$ to the left of $d^{2} \sqrt{2 d}$ .

$\frac{25 d^{2} \sqrt{2 d}}{4} + 7 d^{2} \sqrt{2 d}$