Finite Math Examples

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

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

Step 1

Rewrite the equation as

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

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

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

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

Step 2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 3

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 4

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 5

Multiply

x^{13 \cdot 5}

$x^{13 \cdot 5}$ by

x^{- 8 \cdot - 5}

$x^{- 8 \cdot - 5}$ by adding the exponents.

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Step 5.1

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 5.2

Simplify each term.

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Step 5.2.1

Multiply

13

$13$ by

5

$5$ .

x^{m \cdot 3} = x^{65 - 8 \cdot - 5}

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

Step 5.2.2

Multiply

- 8

$- 8$ by

- 5

$- 5$ .

x^{m \cdot 3} = x^{65 + 40}

$x^{m \cdot 3} = x^{65 + 40}$

x^{m \cdot 3} = x^{65 + 40}

$x^{m \cdot 3} = x^{65 + 40}$

Step 5.3

Add

65

$65$ and

40

$40$ .

x^{m \cdot 3} = x^{105}

$x^{m \cdot 3} = x^{105}$

x^{m \cdot 3} = x^{105}

$x^{m \cdot 3} = x^{105}$

Step 6

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

m \cdot 3 = 105

$m \cdot 3 = 105$

Step 7

Divide each term in

m \cdot 3 = 105

$m \cdot 3 = 105$ by

3

$3$ and simplify.

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Step 7.1

Divide each term in

m \cdot 3 = 105

$m \cdot 3 = 105$ by

3

$3$ .

\frac{m \cdot 3}{3} = \frac{105}{3}

$\frac{m \cdot 3}{3} = \frac{105}{3}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of

3

$3$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{m \cdot 3}{3} = \frac{105}{3}$

Step 7.2.1.2

Divide

$m$ by

$1$ .

$m = \frac{105}{3}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide

$105$ by

$3$ .

$m = 35$