Algebra Examples

${(x^{m})}^{3} = {(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5}$

Step 1

Simplify the left side.

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Step 1.1

Simplify ${({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{m})}^{3}$ .

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Step 1.1.1

Multiply the exponents in ${(x^{13})}^{5}$ .

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Step 1.1.1.1

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

${({(x^{13 \cdot 5} \cdot {(x^{- 8})}^{- 5})}^{m})}^{3} = {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{13})}^{5} \cdot {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$

Step 1.1.1.2

Multiply $13$ by $5$ .

${({(x^{65} \cdot {(x^{- 8})}^{- 5})}^{m})}^{3} = {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{13})}^{5} \cdot {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$

Step 1.1.2

Multiply the exponents in ${(x^{- 8})}^{- 5}$ .

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Step 1.1.2.1

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

${({(x^{65} \cdot x^{- 8 \cdot - 5})}^{m})}^{3} = {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{13})}^{5} \cdot {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$

Step 1.1.2.2

Multiply $- 8$ by $- 5$ .

${({(x^{65} \cdot x^{40})}^{m})}^{3} = {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{13})}^{5} \cdot {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$

Step 1.1.3

Multiply $x^{65}$ by $x^{40}$ by adding the exponents.

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Step 1.1.3.1

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

${({(x^{65 + 40})}^{m})}^{3} = {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{13})}^{5} \cdot {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$

Step 1.1.3.2

Add $65$ and $40$ .

${({(x^{105})}^{m})}^{3} = {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{13})}^{5} \cdot {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$

Step 1.1.4

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

${(x^{105 m})}^{3} = {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{13})}^{5} \cdot {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$

Step 1.1.5

Multiply the exponents in ${(x^{105 m})}^{3}$ .

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Step 1.1.5.1

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

$x^{105 m \cdot 3} = {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{13})}^{5} \cdot {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$

Step 1.1.5.2

Multiply $3$ by $105$ .

$x^{315 m} = {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{13})}^{5} \cdot {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$

Step 2

Simplify the right side.

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Step 2.1

Simplify ${({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{13})}^{5} \cdot {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$ .

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Step 2.1.1

Multiply the exponents in ${({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{13})}^{5}$ .

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Step 2.1.1.1

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

$x^{315 m} = {({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{13 \cdot 5} \cdot {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$

Step 2.1.1.2

Multiply $13$ by $5$ .

$x^{315 m} = {({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{65} \cdot {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$

Step 2.1.2

Multiply the exponents in ${(x^{13})}^{5}$ .

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Step 2.1.2.1

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

$x^{315 m} = {(x^{13 \cdot 5} \cdot {(x^{- 8})}^{- 5})}^{65} \cdot {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$

Step 2.1.2.2

Multiply $13$ by $5$ .

$x^{315 m} = {(x^{65} \cdot {(x^{- 8})}^{- 5})}^{65} \cdot {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$

Step 2.1.3

Multiply the exponents in ${(x^{- 8})}^{- 5}$ .

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Step 2.1.3.1

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

$x^{315 m} = {(x^{65} \cdot x^{- 8 \cdot - 5})}^{65} \cdot {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$

Step 2.1.3.2

Multiply $- 8$ by $- 5$ .

$x^{315 m} = {(x^{65} \cdot x^{40})}^{65} \cdot {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$

Step 2.1.4

Multiply $x^{65}$ by $x^{40}$ by adding the exponents.

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Step 2.1.4.1

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

$x^{315 m} = {(x^{65 + 40})}^{65} \cdot {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$

Step 2.1.4.2

Add $65$ and $40$ .

$x^{315 m} = {(x^{105})}^{65} \cdot {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$

Step 2.1.5

Multiply the exponents in ${(x^{105})}^{65}$ .

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Step 2.1.5.1

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

$x^{315 m} = x^{105 \cdot 65} \cdot {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$

Step 2.1.5.2

Multiply $105$ by $65$ .

$x^{315 m} = x^{6825} \cdot {({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$

Step 2.1.6

Multiply the exponents in ${({({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8})}^{- 5}$ .

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Step 2.1.6.1

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

$x^{315 m} = x^{6825} \cdot {({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{- 8 \cdot - 5}$

Step 2.1.6.2

Multiply $- 8$ by $- 5$ .

$x^{315 m} = x^{6825} \cdot {({(x^{13})}^{5} \cdot {(x^{- 8})}^{- 5})}^{40}$

Step 2.1.7

Multiply the exponents in ${(x^{13})}^{5}$ .

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Step 2.1.7.1

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

$x^{315 m} = x^{6825} \cdot {(x^{13 \cdot 5} \cdot {(x^{- 8})}^{- 5})}^{40}$

Step 2.1.7.2

Multiply $13$ by $5$ .

$x^{315 m} = x^{6825} \cdot {(x^{65} \cdot {(x^{- 8})}^{- 5})}^{40}$

Step 2.1.8

Multiply the exponents in ${(x^{- 8})}^{- 5}$ .

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Step 2.1.8.1

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

$x^{315 m} = x^{6825} \cdot {(x^{65} \cdot x^{- 8 \cdot - 5})}^{40}$

Step 2.1.8.2

Multiply $- 8$ by $- 5$ .

$x^{315 m} = x^{6825} \cdot {(x^{65} \cdot x^{40})}^{40}$

Step 2.1.9

Multiply $x^{65}$ by $x^{40}$ by adding the exponents.

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Step 2.1.9.1

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

$x^{315 m} = x^{6825} \cdot {(x^{65 + 40})}^{40}$

Step 2.1.9.2

Add $65$ and $40$ .

$x^{315 m} = x^{6825} \cdot {(x^{105})}^{40}$

Step 2.1.10

Multiply the exponents in ${(x^{105})}^{40}$ .

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Step 2.1.10.1

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

$x^{315 m} = x^{6825} \cdot x^{105 \cdot 40}$

Step 2.1.10.2

Multiply $105$ by $40$ .

$x^{315 m} = x^{6825} \cdot x^{4200}$

Step 2.1.11

Multiply $x^{6825}$ by $x^{4200}$ by adding the exponents.

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Step 2.1.11.1

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

$x^{315 m} = x^{6825 + 4200}$

Step 2.1.11.2

Add $6825$ and $4200$ .

$x^{315 m} = x^{11025}$