Algebra Examples

${(\frac{1}{7})}^{3 a + 3} = 343^{a - 1}$

Step 1

Apply the product rule to $\frac{1}{7}$ .

$\frac{1^{3 a + 3}}{7^{3 a + 3}} = 343^{a - 1}$

Step 2

One to any power is one.

$\frac{1}{7^{3 a + 3}} = 343^{a - 1}$

Step 3

Move $7^{3 a + 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$7^{- (3 a + 3)} = 343^{a - 1}$

Step 4

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

$7^{- (3 a + 3)} = 7^{3 (a - 1)}$

Step 5

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

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

Step 6

Solve for $a$ .

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Step 6.1

Simplify $- (3 a + 3)$ .

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Step 6.1.1

Rewrite.

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

Step 6.1.2

Simplify by adding zeros.

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

Step 6.1.3

Apply the distributive property.

$- (3 a) - 1 \cdot 3 = 3 (a - 1)$

Step 6.1.4

Multiply.

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Step 6.1.4.1

Multiply $3$ by $- 1$ .

$- 3 a - 1 \cdot 3 = 3 (a - 1)$

Step 6.1.4.2

Multiply $- 1$ by $3$ .

$- 3 a - 3 = 3 (a - 1)$

Step 6.2

Simplify $3 (a - 1)$ .

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Step 6.2.1

Apply the distributive property.

$- 3 a - 3 = 3 a + 3 \cdot - 1$

Step 6.2.2

Multiply $3$ by $- 1$ .

$- 3 a - 3 = 3 a - 3$

Step 6.3

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

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Step 6.3.1

Subtract $3 a$ from both sides of the equation.

$- 3 a - 3 - 3 a = - 3$

Step 6.3.2

Subtract $3 a$ from $- 3 a$ .

$- 6 a - 3 = - 3$

Step 6.4

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

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Step 6.4.1

Add $3$ to both sides of the equation.

$- 6 a = - 3 + 3$

Step 6.4.2

Add $- 3$ and $3$ .

$- 6 a = 0$

Step 6.5

Divide each term in $- 6 a = 0$ by $- 6$ and simplify.

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Step 6.5.1

Divide each term in $- 6 a = 0$ by $- 6$ .

$\frac{- 6 a}{- 6} = \frac{0}{- 6}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $- 6$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{- 6 a}{- 6} = \frac{0}{- 6}$

Step 6.5.2.1.2

Divide $a$ by $1$ .

$a = \frac{0}{- 6}$

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Divide $0$ by $- 6$ .

$a = 0$