Algebra Examples

$\frac{{(4 g^{3} h^{2} k^{4})}^{3}}{8 g^{3} h^{2}} - {(h^{5} k^{3})}^{5}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $4 g^{3} h^{2} k^{4}$ .

$\frac{{(4 g^{3} h^{2})}^{3} {(k^{4})}^{3}}{8 g^{3} h^{2}} - {(h^{5} k^{3})}^{5}$

Step 1.2

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

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Step 1.2.1

Apply the product rule to $4 g^{3} h^{2}$ .

$\frac{{(4 g^{3})}^{3} {(h^{2})}^{3} {(k^{4})}^{3}}{8 g^{3} h^{2}} - {(h^{5} k^{3})}^{5}$

Step 1.2.2

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

$\frac{4^{3} {(g^{3})}^{3} {(h^{2})}^{3} {(k^{4})}^{3}}{8 g^{3} h^{2}} - {(h^{5} k^{3})}^{5}$

Step 1.3

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

$\frac{64 {(g^{3})}^{3} {(h^{2})}^{3} {(k^{4})}^{3}}{8 g^{3} h^{2}} - {(h^{5} k^{3})}^{5}$

Step 1.4

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

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Step 1.4.1

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

$\frac{64 g^{3 \cdot 3} {(h^{2})}^{3} {(k^{4})}^{3}}{8 g^{3} h^{2}} - {(h^{5} k^{3})}^{5}$

Step 1.4.2

Multiply $3$ by $3$ .

$\frac{64 g^{9} {(h^{2})}^{3} {(k^{4})}^{3}}{8 g^{3} h^{2}} - {(h^{5} k^{3})}^{5}$

Step 1.5

Multiply the exponents in ${(h^{2})}^{3}$ .

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Step 1.5.1

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

$\frac{64 g^{9} h^{2 \cdot 3} {(k^{4})}^{3}}{8 g^{3} h^{2}} - {(h^{5} k^{3})}^{5}$

Step 1.5.2

Multiply $2$ by $3$ .

$\frac{64 g^{9} h^{6} {(k^{4})}^{3}}{8 g^{3} h^{2}} - {(h^{5} k^{3})}^{5}$

Step 1.6

Multiply the exponents in ${(k^{4})}^{3}$ .

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Step 1.6.1

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

$\frac{64 g^{9} h^{6} k^{4 \cdot 3}}{8 g^{3} h^{2}} - {(h^{5} k^{3})}^{5}$

Step 1.6.2

Multiply $4$ by $3$ .

$\frac{64 g^{9} h^{6} k^{12}}{8 g^{3} h^{2}} - {(h^{5} k^{3})}^{5}$

Step 2

Cancel the common factor of $64$ and $8$ .

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Step 2.1

Factor $8$ out of $64 g^{9} h^{6} k^{12}$ .

$\frac{8 (8 g^{9} h^{6} k^{12})}{8 g^{3} h^{2}} - {(h^{5} k^{3})}^{5}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $8$ out of $8 g^{3} h^{2}$ .

$\frac{8 (8 g^{9} h^{6} k^{12})}{8 (g^{3} h^{2})} - {(h^{5} k^{3})}^{5}$

Step 2.2.2

Cancel the common factor.

$\frac{8 (8 g^{9} h^{6} k^{12})}{8 (g^{3} h^{2})} - {(h^{5} k^{3})}^{5}$

Step 2.2.3

Rewrite the expression.

$\frac{8 g^{9} h^{6} k^{12}}{g^{3} h^{2}} - {(h^{5} k^{3})}^{5}$

Step 3

Cancel the common factor of $g^{9}$ and $g^{3}$ .

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Step 3.1

Factor $g^{3}$ out of $8 g^{9} h^{6} k^{12}$ .

$\frac{g^{3} (8 g^{6} h^{6} k^{12})}{g^{3} h^{2}} - {(h^{5} k^{3})}^{5}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $g^{3}$ out of $g^{3} h^{2}$ .

$\frac{g^{3} (8 g^{6} h^{6} k^{12})}{g^{3} (h^{2})} - {(h^{5} k^{3})}^{5}$

Step 3.2.2

Cancel the common factor.

$\frac{g^{3} (8 g^{6} h^{6} k^{12})}{g^{3} h^{2}} - {(h^{5} k^{3})}^{5}$

Step 3.2.3

Rewrite the expression.

$\frac{8 g^{6} h^{6} k^{12}}{h^{2}} - {(h^{5} k^{3})}^{5}$

Step 4

Cancel the common factor of $h^{6}$ and $h^{2}$ .

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Step 4.1

Factor $h^{2}$ out of $8 g^{6} h^{6} k^{12}$ .

$\frac{h^{2} (8 g^{6} h^{4} k^{12})}{h^{2}} - {(h^{5} k^{3})}^{5}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Multiply by $1$ .

$\frac{h^{2} (8 g^{6} h^{4} k^{12})}{h^{2} \cdot 1} - {(h^{5} k^{3})}^{5}$

Step 4.2.2

Cancel the common factor.

$\frac{h^{2} (8 g^{6} h^{4} k^{12})}{h^{2} \cdot 1} - {(h^{5} k^{3})}^{5}$

Step 4.2.3

Rewrite the expression.

$\frac{8 g^{6} h^{4} k^{12}}{1} - {(h^{5} k^{3})}^{5}$

Step 4.2.4

Divide $8 g^{6} h^{4} k^{12}$ by $1$ .

$8 g^{6} h^{4} k^{12} - {(h^{5} k^{3})}^{5}$

Step 5

Apply the product rule to $h^{5} k^{3}$ .

$8 g^{6} h^{4} k^{12} - ({(h^{5})}^{5} {(k^{3})}^{5})$

Step 6

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

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Step 6.1

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

$8 g^{6} h^{4} k^{12} - (h^{5 \cdot 5} {(k^{3})}^{5})$

Step 6.2

Multiply $5$ by $5$ .

$8 g^{6} h^{4} k^{12} - (h^{25} {(k^{3})}^{5})$

Step 7

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

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Step 7.1

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

$8 g^{6} h^{4} k^{12} - (h^{25} k^{3 \cdot 5})$

Step 7.2

Multiply $3$ by $5$ .

$8 g^{6} h^{4} k^{12} - (h^{25} k^{15})$

$8 g^{6} h^{4} k^{12} - h^{25} k^{15}$