Algebra Examples

$8^{2 a - 1} = 32^{2 a + 1}$

Step 1

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

$2^{3 (2 a - 1)} = 2^{5 (2 a + 1)}$

Step 2

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

$3 (2 a - 1) = 5 (2 a + 1)$

Step 3

Solve for $a$ .

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Step 3.1

Simplify $3 (2 a - 1)$ .

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Step 3.1.1

Rewrite.

$0 + 0 + 3 (2 a - 1) = 5 (2 a + 1)$

Step 3.1.2

Simplify by adding zeros.

$3 (2 a - 1) = 5 (2 a + 1)$

Step 3.1.3

Apply the distributive property.

$3 (2 a) + 3 \cdot - 1 = 5 (2 a + 1)$

Step 3.1.4

Multiply.

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Step 3.1.4.1

Multiply $2$ by $3$ .

$6 a + 3 \cdot - 1 = 5 (2 a + 1)$

Step 3.1.4.2

Multiply $3$ by $- 1$ .

$6 a - 3 = 5 (2 a + 1)$

Step 3.2

Simplify $5 (2 a + 1)$ .

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Step 3.2.1

Apply the distributive property.

$6 a - 3 = 5 (2 a) + 5 \cdot 1$

Step 3.2.2

Multiply.

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Step 3.2.2.1

Multiply $2$ by $5$ .

$6 a - 3 = 10 a + 5 \cdot 1$

Step 3.2.2.2

Multiply $5$ by $1$ .

$6 a - 3 = 10 a + 5$

Step 3.3

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

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Step 3.3.1

Subtract $10 a$ from both sides of the equation.

$6 a - 3 - 10 a = 5$

Step 3.3.2

Subtract $10 a$ from $6 a$ .

$- 4 a - 3 = 5$

Step 3.4

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

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Step 3.4.1

Add $3$ to both sides of the equation.

$- 4 a = 5 + 3$

Step 3.4.2

Add $5$ and $3$ .

$- 4 a = 8$

Step 3.5

Divide each term in $- 4 a = 8$ by $- 4$ and simplify.

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Step 3.5.1

Divide each term in $- 4 a = 8$ by $- 4$ .

$\frac{- 4 a}{- 4} = \frac{8}{- 4}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $- 4$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{- 4 a}{- 4} = \frac{8}{- 4}$

Step 3.5.2.1.2

Divide $a$ by $1$ .

$a = \frac{8}{- 4}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Divide $8$ by $- 4$ .

$a = - 2$