Algebra Examples

$125^{- x - 7} \cdot 5^{6 x - 12} = 125^{2 x + 3}$

Step 1

Rewrite $125$ as $5^{3}$ .

${(5^{3})}^{- x - 7} \cdot 5^{6 x - 12} = 125^{2 x + 3}$

Step 2

Multiply the exponents in ${(5^{3})}^{- x - 7}$ .

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Step 2.1

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

$5^{3 (- x - 7)} \cdot 5^{6 x - 12} = 125^{2 x + 3}$

Step 2.2

Apply the distributive property.

$5^{3 (- x) + 3 \cdot - 7} \cdot 5^{6 x - 12} = 125^{2 x + 3}$

Step 2.3

Multiply $- 1$ by $3$ .

$5^{- 3 x + 3 \cdot - 7} \cdot 5^{6 x - 12} = 125^{2 x + 3}$

Step 2.4

Multiply $3$ by $- 7$ .

$5^{- 3 x - 21} \cdot 5^{6 x - 12} = 125^{2 x + 3}$

Step 3

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

$5^{- 3 x - 21 + 6 x - 12} = 125^{2 x + 3}$

Step 4

Add $- 3 x$ and $6 x$ .

$5^{3 x - 21 - 12} = 125^{2 x + 3}$

Step 5

Subtract $12$ from $- 21$ .

$5^{3 x - 33} = 125^{2 x + 3}$

Step 6

Create equivalent expressions in the equation that all have equal bases.

$5^{3 x - 33} = 5^{3 (2 x + 3)}$

Step 7

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

$3 x - 33 = 3 (2 x + 3)$

Step 8

Solve for $x$ .

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Step 8.1

Simplify $3 (2 x + 3)$ .

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Step 8.1.1

Rewrite.

$3 x - 33 = 0 + 0 + 3 (2 x + 3)$

Step 8.1.2

Simplify by adding zeros.

$3 x - 33 = 3 (2 x + 3)$

Step 8.1.3

Apply the distributive property.

$3 x - 33 = 3 (2 x) + 3 \cdot 3$

Step 8.1.4

Multiply.

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Step 8.1.4.1

Multiply $2$ by $3$ .

$3 x - 33 = 6 x + 3 \cdot 3$

Step 8.1.4.2

Multiply $3$ by $3$ .

$3 x - 33 = 6 x + 9$

Step 8.2

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

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Step 8.2.1

Subtract $6 x$ from both sides of the equation.

$3 x - 33 - 6 x = 9$

Step 8.2.2

Subtract $6 x$ from $3 x$ .

$- 3 x - 33 = 9$

Step 8.3

Move all terms not containing $x$ to the right side of the equation.

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Step 8.3.1

Add $33$ to both sides of the equation.

$- 3 x = 9 + 33$

Step 8.3.2

Add $9$ and $33$ .

$- 3 x = 42$

Step 8.4

Divide each term in $- 3 x = 42$ by $- 3$ and simplify.

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Step 8.4.1

Divide each term in $- 3 x = 42$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{42}{- 3}$

Step 8.4.2

Simplify the left side.

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Step 8.4.2.1

Cancel the common factor of $- 3$ .

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Step 8.4.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{42}{- 3}$

Step 8.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{42}{- 3}$

Step 8.4.3

Simplify the right side.

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Step 8.4.3.1

Divide $42$ by $- 3$ .

$x = - 14$