Algebra Examples

${(\frac{3}{2} d^{- 2} f^{4})}^{4} {(- \frac{4}{3} d^{5} f)}^{3}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{3}{2} \cdot (\frac{1}{d^{2}} f^{4}))}^{4} {(- \frac{4}{3} \cdot (d^{5} f))}^{3}$

Step 2

Combine $\frac{1}{d^{2}}$ and $f^{4}$ .

${(\frac{3}{2} \cdot \frac{f^{4}}{d^{2}})}^{4} {(- \frac{4}{3} \cdot (d^{5} f))}^{3}$

Step 3

Combine.

${(\frac{3 f^{4}}{2 d^{2}})}^{4} {(- \frac{4}{3} \cdot (d^{5} f))}^{3}$

Step 4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1

Apply the product rule to $\frac{3 f^{4}}{2 d^{2}}$ .

$\frac{{(3 f^{4})}^{4}}{{(2 d^{2})}^{4}} {(- \frac{4}{3} \cdot (d^{5} f))}^{3}$

Step 4.2

Apply the product rule to $3 f^{4}$ .

$\frac{3^{4} {(f^{4})}^{4}}{{(2 d^{2})}^{4}} {(- \frac{4}{3} \cdot (d^{5} f))}^{3}$

Step 4.3

Apply the product rule to $2 d^{2}$ .

$\frac{3^{4} {(f^{4})}^{4}}{2^{4} {(d^{2})}^{4}} {(- \frac{4}{3} \cdot (d^{5} f))}^{3}$

Step 5

Simplify the numerator.

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Step 5.1

Raise $3$ to the power of $4$ .

$\frac{81 {(f^{4})}^{4}}{2^{4} {(d^{2})}^{4}} {(- \frac{4}{3} \cdot (d^{5} f))}^{3}$

Step 5.2

Multiply the exponents in ${(f^{4})}^{4}$ .

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Step 5.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{81 f^{4 \cdot 4}}{2^{4} {(d^{2})}^{4}} {(- \frac{4}{3} \cdot (d^{5} f))}^{3}$

Step 5.2.2

Multiply $4$ by $4$ .

$\frac{81 f^{16}}{2^{4} {(d^{2})}^{4}} {(- \frac{4}{3} \cdot (d^{5} f))}^{3}$

Step 6

Simplify the denominator.

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Step 6.1

Raise $2$ to the power of $4$ .

$\frac{81 f^{16}}{16 {(d^{2})}^{4}} {(- \frac{4}{3} \cdot (d^{5} f))}^{3}$

Step 6.2

Multiply the exponents in ${(d^{2})}^{4}$ .

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Step 6.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{81 f^{16}}{16 d^{2 \cdot 4}} {(- \frac{4}{3} \cdot (d^{5} f))}^{3}$

Step 6.2.2

Multiply $2$ by $4$ .

$\frac{81 f^{16}}{16 d^{8}} {(- \frac{4}{3} \cdot (d^{5} f))}^{3}$

Step 7

Multiply $- \frac{4}{3} (d^{5} f)$ .

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Step 7.1

Combine $d^{5}$ and $\frac{4}{3}$ .

$\frac{81 f^{16}}{16 d^{8}} {(- \frac{d^{5} \cdot 4}{3} f)}^{3}$

Step 7.2

Combine $f$ and $\frac{d^{5} \cdot 4}{3}$ .

$\frac{81 f^{16}}{16 d^{8}} {(- \frac{f (d^{5} \cdot 4)}{3})}^{3}$

Step 8

Move $4$ to the left of $f d^{5}$ .

$\frac{81 f^{16}}{16 d^{8}} {(- \frac{4 \cdot (f d^{5})}{3})}^{3}$

Step 9

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 9.1

Apply the product rule to $- \frac{4 f d^{5}}{3}$ .

$\frac{81 f^{16}}{16 d^{8}} ({(- 1)}^{3} {(\frac{4 f d^{5}}{3})}^{3})$

Step 9.2

Apply the product rule to $\frac{4 f d^{5}}{3}$ .

$\frac{81 f^{16}}{16 d^{8}} ({(- 1)}^{3} \frac{{(4 f d^{5})}^{3}}{3^{3}})$

Step 9.3

Apply the product rule to $4 f d^{5}$ .

$\frac{81 f^{16}}{16 d^{8}} ({(- 1)}^{3} \frac{{(4 f)}^{3} {(d^{5})}^{3}}{3^{3}})$

Step 9.4

Apply the product rule to $4 f$ .

$\frac{81 f^{16}}{16 d^{8}} ({(- 1)}^{3} \frac{4^{3} f^{3} {(d^{5})}^{3}}{3^{3}})$

Step 10

Simplify the expression.

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Step 10.1

Rewrite using the commutative property of multiplication.

${(- 1)}^{3} \frac{81 f^{16}}{16 d^{8}} \frac{4^{3} f^{3} {(d^{5})}^{3}}{3^{3}}$

Step 10.2

Raise $- 1$ to the power of $3$ .

$- \frac{81 f^{16}}{16 d^{8}} \cdot \frac{4^{3} f^{3} {(d^{5})}^{3}}{3^{3}}$

Step 11

Simplify the numerator.

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Step 11.1

Raise $4$ to the power of $3$ .

$- \frac{81 f^{16}}{16 d^{8}} \cdot \frac{64 f^{3} {(d^{5})}^{3}}{3^{3}}$

Step 11.2

Multiply the exponents in ${(d^{5})}^{3}$ .

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Step 11.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{81 f^{16}}{16 d^{8}} \cdot \frac{64 f^{3} d^{5 \cdot 3}}{3^{3}}$

Step 11.2.2

Multiply $5$ by $3$ .

$- \frac{81 f^{16}}{16 d^{8}} \cdot \frac{64 f^{3} d^{15}}{3^{3}}$

Step 12

Reduce the expression by cancelling the common factors.

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Step 12.1

Raise $3$ to the power of $3$ .

$- \frac{81 f^{16}}{16 d^{8}} \cdot \frac{64 f^{3} d^{15}}{27}$

Step 12.2

Cancel the common factor of $27$ .

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Step 12.2.1

Move the leading negative in $- \frac{81 f^{16}}{16 d^{8}}$ into the numerator.

$\frac{- 81 f^{16}}{16 d^{8}} \cdot \frac{64 f^{3} d^{15}}{27}$

Step 12.2.2

Factor $27$ out of $- 81 f^{16}$ .

$\frac{27 (- 3 f^{16})}{16 d^{8}} \cdot \frac{64 f^{3} d^{15}}{27}$

Step 12.2.3

Cancel the common factor.

$\frac{27 (- 3 f^{16})}{16 d^{8}} \cdot \frac{64 f^{3} d^{15}}{27}$

Step 12.2.4

Rewrite the expression.

$\frac{- 3 f^{16}}{16 d^{8}} (64 f^{3} d^{15})$

Step 12.3

Cancel the common factor of $16 d^{8}$ .

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Step 12.3.1

Factor $16 d^{8}$ out of $64 f^{3} d^{15}$ .

$\frac{- 3 f^{16}}{16 d^{8}} (16 d^{8} (4 f^{3} d^{7}))$

Step 12.3.2

Cancel the common factor.

$\frac{- 3 f^{16}}{16 d^{8}} (16 d^{8} (4 f^{3} d^{7}))$

Step 12.3.3

Rewrite the expression.

$- 3 f^{16} (4 f^{3} d^{7})$

Step 12.4

Multiply $4$ by $- 3$ .

$- 12 f^{16} (f^{3} d^{7})$

Step 13

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

$- 12 f^{3 + 16} d^{7}$

Step 14

Add $3$ and $16$ .

$- 12 f^{19} d^{7}$