Algebra Examples

$6^{3 m} \cdot 6^{- m} = 6^{- 2 m}$

Step 1

Subtract $6^{- 2 m}$ from both sides of the equation.

$6^{3 m} \cdot 6^{- m} - 6^{- 2 m} = 0$

Step 2

Multiply $6^{3 m}$ by $6^{- m}$ by adding the exponents.

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Step 2.1

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

$6^{3 m - m} - 6^{- 2 m} = 0$

Step 2.2

Subtract $m$ from $3 m$ .

$6^{2 m} - 6^{- 2 m} = 0$

Step 3

Rewrite $6^{2 m}$ as ${(6^{m})}^{2}$ .

${(6^{m})}^{2} - 6^{- 2 m} = 0$

Step 4

Rewrite $6^{- 2 m}$ as ${(6^{- m})}^{2}$ .

${(6^{m})}^{2} - {(6^{- m})}^{2} = 0$

Step 5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6^{m}$ and $b = 6^{- m}$ .

$(6^{m} + 6^{- m}) (6^{m} - 6^{- m}) = 0$

Step 6

Simplify $(6^{m} + 6^{- m}) (6^{m} - 6^{- m})$ .

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Step 6.1

Expand $(6^{m} + 6^{- m}) (6^{m} - 6^{- m})$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$6^{m} (6^{m} - 6^{- m}) + 6^{- m} (6^{m} - 6^{- m}) = 0$

Step 6.1.2

Apply the distributive property.

$6^{m} \cdot 6^{m} + 6^{m} (- 6^{- m}) + 6^{- m} (6^{m} - 6^{- m}) = 0$

Step 6.1.3

Apply the distributive property.

$6^{m} \cdot 6^{m} + 6^{m} (- 6^{- m}) + 6^{- m} \cdot 6^{m} + 6^{- m} (- 6^{- m}) = 0$

Step 6.2

Simplify terms.

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Step 6.2.1

Combine the opposite terms in $6^{m} \cdot 6^{m} + 6^{m} (- 6^{- m}) + 6^{- m} \cdot 6^{m} + 6^{- m} (- 6^{- m})$ .

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Step 6.2.1.1

Reorder the factors in the terms $6^{m} (- 6^{- m})$ and $6^{- m} \cdot 6^{m}$ .

$6^{m} \cdot 6^{m} - 6^{- m} \cdot 6^{m} + 6^{- m} \cdot 6^{m} + 6^{- m} (- 6^{- m}) = 0$

Step 6.2.1.2

Add $- 6^{- m} \cdot 6^{m}$ and $6^{- m} \cdot 6^{m}$ .

$6^{m} \cdot 6^{m} + 0 + 6^{- m} (- 6^{- m}) = 0$

Step 6.2.1.3

Add $6^{m} \cdot 6^{m}$ and $0$ .

$6^{m} \cdot 6^{m} + 6^{- m} (- 6^{- m}) = 0$

Step 6.2.2

Simplify each term.

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Step 6.2.2.1

Multiply $6^{m}$ by $6^{m}$ by adding the exponents.

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Step 6.2.2.1.1

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

$6^{m + m} + 6^{- m} (- 6^{- m}) = 0$

Step 6.2.2.1.2

Add $m$ and $m$ .

$6^{2 m} + 6^{- m} (- 6^{- m}) = 0$

Step 6.2.2.2

Rewrite using the commutative property of multiplication.

$6^{2 m} - 6^{- m} \cdot 6^{- m} = 0$

Step 6.2.2.3

Multiply $6^{- m}$ by $6^{- m}$ by adding the exponents.

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Step 6.2.2.3.1

Move $6^{- m}$ .

$6^{2 m} - (6^{- m} \cdot 6^{- m}) = 0$

Step 6.2.2.3.2

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

$6^{2 m} - 6^{- m - m} = 0$

Step 6.2.2.3.3

Subtract $m$ from $- m$ .

$6^{2 m} - 6^{- 2 m} = 0$

Step 7

Move $- 6^{- 2 m}$ to the right side of the equation by adding it to both sides.

$6^{2 m} = 6^{- 2 m}$

Step 8

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

$2 m = - 2 m$

Step 9

Solve for $m$ .

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Step 9.1

Move all terms containing $m$ to the left side of the equation.

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Step 9.1.1

Add $2 m$ to both sides of the equation.

$2 m + 2 m = 0$

Step 9.1.2

Add $2 m$ and $2 m$ .

$4 m = 0$

Step 9.2

Divide each term in $4 m = 0$ by $4$ and simplify.

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Step 9.2.1

Divide each term in $4 m = 0$ by $4$ .

$\frac{4 m}{4} = \frac{0}{4}$

Step 9.2.2

Simplify the left side.

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Step 9.2.2.1

Cancel the common factor of $4$ .

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Step 9.2.2.1.1

Cancel the common factor.

$\frac{4 m}{4} = \frac{0}{4}$

Step 9.2.2.1.2

Divide $m$ by $1$ .

$m = \frac{0}{4}$

Step 9.2.3

Simplify the right side.

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Step 9.2.3.1

Divide $0$ by $4$ .

$m = 0$