Trigonometry Examples

$\frac{1}{25^{3 x}} = 625^{3 x}$

Step 1

Move $25^{3 x}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$25^{- 3 x} = 625^{3 x}$

Step 2

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

$5^{2 (- 3 x)} = 5^{4 (3 x)}$

Step 3

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

$2 (- 3 x) = 4 (3 x)$

Step 4

Solve for $x$ .

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Step 4.1

Multiply $- 3$ by $2$ .

$- 6 x = 4 \cdot (3 x)$

Step 4.2

Multiply $3$ by $4$ .

$- 6 x = 12 x$

Step 4.3

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

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Step 4.3.1

Subtract $12 x$ from both sides of the equation.

$- 6 x - 12 x = 0$

Step 4.3.2

Subtract $12 x$ from $- 6 x$ .

$- 18 x = 0$

Step 4.4

Divide each term in $- 18 x = 0$ by $- 18$ and simplify.

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Step 4.4.1

Divide each term in $- 18 x = 0$ by $- 18$ .

$\frac{- 18 x}{- 18} = \frac{0}{- 18}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $- 18$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{- 18 x}{- 18} = \frac{0}{- 18}$

Step 4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{- 18}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Divide $0$ by $- 18$ .

$x = 0$