Algebra Examples

$243^{- y} = {(\frac{1}{243})}^{3 y} \cdot 9^{- 2 y}$

Step 1

Apply the product rule to $\frac{1}{243}$ .

$243^{- y} = \frac{1^{3 y}}{243^{3 y}} \cdot 9^{- 2 y}$

Step 2

One to any power is one.

$243^{- y} = \frac{1}{243^{3 y}} \cdot 9^{- 2 y}$

Step 3

Move $243^{3 y}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$243^{- y} = 243^{- 3 y} \cdot 9^{- 2 y}$

Step 4

Rewrite $243$ as $3^{5}$ .

$243^{- y} = {(3^{5})}^{- 3 y} \cdot 9^{- 2 y}$

Step 5

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

Tap for more steps...

Step 5.1

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

$243^{- y} = 3^{5 (- 3 y)} \cdot 9^{- 2 y}$

Step 5.2

Multiply $- 3$ by $5$ .

$243^{- y} = 3^{- 15 y} \cdot 9^{- 2 y}$

Step 6

Rewrite $9$ as $3^{2}$ .

$243^{- y} = 3^{- 15 y} \cdot {(3^{2})}^{- 2 y}$

Step 7

Multiply the exponents in ${(3^{2})}^{- 2 y}$ .

Tap for more steps...

Step 7.1

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

$243^{- y} = 3^{- 15 y} \cdot 3^{2 (- 2 y)}$

Step 7.2

Multiply $- 2$ by $2$ .

$243^{- y} = 3^{- 15 y} \cdot 3^{- 4 y}$

Step 8

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

$243^{- y} = 3^{- 15 y - 4 y}$

Step 9

Subtract $4 y$ from $- 15 y$ .

$243^{- y} = 3^{- 19 y}$

Step 10

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

$3^{5 (- y)} = 3^{- 19 y}$

Step 11

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

$5 (- y) = - 19 y$

Step 12

Solve for $y$ .

Tap for more steps...

Step 12.1

Multiply $- 1$ by $5$ .

$- 5 y = - 19 y$

Step 12.2

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

Tap for more steps...

Step 12.2.1

Add $19 y$ to both sides of the equation.

$- 5 y + 19 y = 0$

Step 12.2.2

Add $- 5 y$ and $19 y$ .

$14 y = 0$

Step 12.3

Divide each term in $14 y = 0$ by $14$ and simplify.

Tap for more steps...

Step 12.3.1

Divide each term in $14 y = 0$ by $14$ .

$\frac{14 y}{14} = \frac{0}{14}$

Step 12.3.2

Simplify the left side.

Tap for more steps...

Step 12.3.2.1

Cancel the common factor of $14$ .

Tap for more steps...

Step 12.3.2.1.1

Cancel the common factor.

$\frac{14 y}{14} = \frac{0}{14}$

Step 12.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{0}{14}$

Step 12.3.3

Simplify the right side.

Tap for more steps...

Step 12.3.3.1

Divide $0$ by $14$ .

$y = 0$