Algebra Examples

$(7 d^{\frac{3}{2}} \cdot 2 g^{\frac{5}{6}}) (2 g^{\frac{3}{2}} \cdot 7 d^{\frac{5}{6}})$

Step 1

Multiply $d^{\frac{3}{2}}$ by $d^{\frac{5}{6}}$ by adding the exponents.

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Step 1.1

Move $d^{\frac{5}{6}}$ .

$7 (d^{\frac{5}{6}} d^{\frac{3}{2}}) \cdot (2 g^{\frac{5}{6}}) (2 g^{\frac{3}{2}} \cdot (7))$

Step 1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$7 d^{\frac{5}{6} + \frac{3}{2}} \cdot (2 g^{\frac{5}{6}}) (2 g^{\frac{3}{2}} \cdot (7))$

Step 1.3

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

$7 d^{\frac{5}{6} + \frac{3}{2} \cdot \frac{3}{3}} \cdot (2 g^{\frac{5}{6}}) (2 g^{\frac{3}{2}} \cdot (7))$

Step 1.4

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

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Step 1.4.1

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

$7 d^{\frac{5}{6} + \frac{3 \cdot 3}{2 \cdot 3}} \cdot (2 g^{\frac{5}{6}}) (2 g^{\frac{3}{2}} \cdot (7))$

Step 1.4.2

Multiply $2$ by $3$ .

$7 d^{\frac{5}{6} + \frac{3 \cdot 3}{6}} \cdot (2 g^{\frac{5}{6}}) (2 g^{\frac{3}{2}} \cdot (7))$

Step 1.5

Combine the numerators over the common denominator.

$7 d^{\frac{5 + 3 \cdot 3}{6}} \cdot (2 g^{\frac{5}{6}}) (2 g^{\frac{3}{2}} \cdot (7))$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $3$ by $3$ .

$7 d^{\frac{5 + 9}{6}} \cdot (2 g^{\frac{5}{6}}) (2 g^{\frac{3}{2}} \cdot (7))$

Step 1.6.2

Add $5$ and $9$ .

$7 d^{\frac{14}{6}} \cdot (2 g^{\frac{5}{6}}) (2 g^{\frac{3}{2}} \cdot (7))$

Step 1.7

Cancel the common factor of $14$ and $6$ .

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Step 1.7.1

Factor $2$ out of $14$ .

$7 d^{\frac{2 (7)}{6}} \cdot (2 g^{\frac{5}{6}}) (2 g^{\frac{3}{2}} \cdot (7))$

Step 1.7.2

Cancel the common factors.

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Step 1.7.2.1

Factor $2$ out of $6$ .

$7 d^{\frac{2 \cdot 7}{2 \cdot 3}} \cdot (2 g^{\frac{5}{6}}) (2 g^{\frac{3}{2}} \cdot (7))$

Step 1.7.2.2

Cancel the common factor.

$7 d^{\frac{2 \cdot 7}{2 \cdot 3}} \cdot (2 g^{\frac{5}{6}}) (2 g^{\frac{3}{2}} \cdot (7))$

Step 1.7.2.3

Rewrite the expression.

$7 d^{\frac{7}{3}} \cdot (2 g^{\frac{5}{6}}) (2 g^{\frac{3}{2}} \cdot (7))$

Step 2

Multiply $g^{\frac{5}{6}}$ by $g^{\frac{3}{2}}$ by adding the exponents.

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Step 2.1

Move $g^{\frac{3}{2}}$ .

$7 d^{\frac{7}{3}} \cdot (2 (g^{\frac{3}{2}} g^{\frac{5}{6}})) (2 \cdot (7))$

Step 2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$7 d^{\frac{7}{3}} \cdot (2 g^{\frac{3}{2} + \frac{5}{6}}) (2 \cdot (7))$

Step 2.3

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

$7 d^{\frac{7}{3}} \cdot (2 g^{\frac{3}{2} \cdot \frac{3}{3} + \frac{5}{6}}) (2 \cdot (7))$

Step 2.4

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

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Step 2.4.1

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

$7 d^{\frac{7}{3}} \cdot (2 g^{\frac{3 \cdot 3}{2 \cdot 3} + \frac{5}{6}}) (2 \cdot (7))$

Step 2.4.2

Multiply $2$ by $3$ .

$7 d^{\frac{7}{3}} \cdot (2 g^{\frac{3 \cdot 3}{6} + \frac{5}{6}}) (2 \cdot (7))$

Step 2.5

Combine the numerators over the common denominator.

$7 d^{\frac{7}{3}} \cdot (2 g^{\frac{3 \cdot 3 + 5}{6}}) (2 \cdot (7))$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $3$ by $3$ .

$7 d^{\frac{7}{3}} \cdot (2 g^{\frac{9 + 5}{6}}) (2 \cdot (7))$

Step 2.6.2

Add $9$ and $5$ .

$7 d^{\frac{7}{3}} \cdot (2 g^{\frac{14}{6}}) (2 \cdot (7))$

Step 2.7

Cancel the common factor of $14$ and $6$ .

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Step 2.7.1

Factor $2$ out of $14$ .

$7 d^{\frac{7}{3}} \cdot (2 g^{\frac{2 (7)}{6}}) (2 \cdot (7))$

Step 2.7.2

Cancel the common factors.

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Step 2.7.2.1

Factor $2$ out of $6$ .

$7 d^{\frac{7}{3}} \cdot (2 g^{\frac{2 \cdot 7}{2 \cdot 3}}) (2 \cdot (7))$

Step 2.7.2.2

Cancel the common factor.

$7 d^{\frac{7}{3}} \cdot (2 g^{\frac{2 \cdot 7}{2 \cdot 3}}) (2 \cdot (7))$

Step 2.7.2.3

Rewrite the expression.

$7 d^{\frac{7}{3}} \cdot (2 g^{\frac{7}{3}}) (2 \cdot (7))$

Step 3

Rewrite using the commutative property of multiplication.

$7 \cdot 2 d^{\frac{7}{3}} g^{\frac{7}{3}} (2 \cdot (7))$

Step 4

Multiply $7$ by $2$ .

$14 d^{\frac{7}{3}} g^{\frac{7}{3}} (2 \cdot (7))$

Step 5

Multiply $2$ by $7$ .

$14 d^{\frac{7}{3}} g^{\frac{7}{3}} \cdot 14$

Step 6

Multiply $14$ by $14$ .

$196 d^{\frac{7}{3}} g^{\frac{7}{3}}$