Algebra Examples

$25^{2 - 2 x} = {(\frac{1}{625})}^{x + 1}$

Step 1

Apply the product rule to $\frac{1}{625}$ .

$25^{2 - 2 x} = \frac{1^{x + 1}}{625^{x + 1}}$

Step 2

One to any power is one.

$25^{2 - 2 x} = \frac{1}{625^{x + 1}}$

Step 3

Move $625^{x + 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$25^{2 - 2 x} = 625^{- (x + 1)}$

Step 4

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

$5^{2 (2 - 2 x)} = 5^{4 (- (x + 1))}$

Step 5

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

$2 (2 - 2 x) = 4 (- (x + 1))$

Step 6

Solve for $x$ .

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Step 6.1

Simplify $2 (2 - 2 x)$ .

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Step 6.1.1

Rewrite.

$0 + 0 + 2 (2 - 2 x) = 4 (- (x + 1))$

Step 6.1.2

Simplify by adding zeros.

$2 (2 - 2 x) = 4 (- (x + 1))$

Step 6.1.3

Apply the distributive property.

$2 \cdot 2 + 2 (- 2 x) = 4 (- (x + 1))$

Step 6.1.4

Multiply.

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Step 6.1.4.1

Multiply $2$ by $2$ .

$4 + 2 (- 2 x) = 4 (- (x + 1))$

Step 6.1.4.2

Multiply $- 2$ by $2$ .

$4 - 4 x = 4 (- (x + 1))$

Step 6.2

Simplify $4 (- (x + 1))$ .

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Step 6.2.1

Apply the distributive property.

$4 - 4 x = 4 (- x - 1 \cdot 1)$

Step 6.2.2

Multiply $- 1$ by $1$ .

$4 - 4 x = 4 (- x - 1)$

Step 6.2.3

Apply the distributive property.

$4 - 4 x = 4 (- x) + 4 \cdot - 1$

Step 6.2.4

Multiply.

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Step 6.2.4.1

Multiply $- 1$ by $4$ .

$4 - 4 x = - 4 x + 4 \cdot - 1$

Step 6.2.4.2

Multiply $4$ by $- 1$ .

$4 - 4 x = - 4 x - 4$

Step 6.3

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

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Step 6.3.1

Add $4 x$ to both sides of the equation.

$4 - 4 x + 4 x = - 4$

Step 6.3.2

Combine the opposite terms in $4 - 4 x + 4 x$ .

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Step 6.3.2.1

Add $- 4 x$ and $4 x$ .

$4 + 0 = - 4$

Step 6.3.2.2

Add $4$ and $0$ .

$4 = - 4$

Step 6.4

Since $4 \neq - 4$ , there are no solutions.

No solution