Finite Math Examples

$3^{\frac{2}{y}} = \frac{1}{81}$

Step 1

Raise $81$ to the power of $1$ .

$3^{\frac{2}{y}} = \frac{1}{81^{1}}$

Step 2

Move $81^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$3^{\frac{2}{y}} = 81^{- 1}$

Step 3

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

$3^{\frac{2}{y}} = 3^{- 4}$

Step 4

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

$\frac{2}{y} = - 4$

Step 5

Solve for $y$ .

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Step 5.1

Find the LCD of the terms in the equation.

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Step 5.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y, 1$

Step 5.1.2

The LCM of one and any expression is the expression.

$y$

Step 5.2

Multiply each term in $\frac{2}{y} = - 4$ by $y$ to eliminate the fractions.

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Step 5.2.1

Multiply each term in $\frac{2}{y} = - 4$ by $y$ .

$\frac{2}{y} y = - 4 y$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $y$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{2}{y} y = - 4 y$

Step 5.2.2.1.2

Rewrite the expression.

$2 = - 4 y$

Step 5.3

Solve the equation.

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Step 5.3.1

Rewrite the equation as $- 4 y = 2$ .

$- 4 y = 2$

Step 5.3.2

Divide each term in $- 4 y = 2$ by $- 4$ and simplify.

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Step 5.3.2.1

Divide each term in $- 4 y = 2$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{2}{- 4}$

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Cancel the common factor of $- 4$ .

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Step 5.3.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{2}{- 4}$

Step 5.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{2}{- 4}$

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Cancel the common factor of $2$ and $- 4$ .

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Step 5.3.2.3.1.1

Factor $2$ out of $2$ .

$y = \frac{2 (1)}{- 4}$

Step 5.3.2.3.1.2

Cancel the common factors.

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Step 5.3.2.3.1.2.1

Factor $2$ out of $- 4$ .

$y = \frac{2 \cdot 1}{2 \cdot - 2}$

Step 5.3.2.3.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot 1}{2 \cdot - 2}$

Step 5.3.2.3.1.2.3

Rewrite the expression.

$y = \frac{1}{- 2}$

Step 5.3.2.3.2

Move the negative in front of the fraction.

$y = - \frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{1}{2}$

Decimal Form:

$y = - 0.5$