Finite Math Examples

$p^{1.3}$

Step 1

Interchange the variables.

$p = y^{1.3}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $y^{1.3} = p$ .

$y^{1.3} = p$

Step 2.2

Convert the decimal exponent to a fractional exponent.

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Step 2.2.1

Convert the decimal number to a fraction by placing the decimal number over a power of ten. Since there is $1$ number to the right of the decimal point, place the decimal number over $10^{1}$ $(10)$ . Next, add the whole number to the left of the decimal.

$y^{1 \frac{3}{10}} = p$

Step 2.2.2

Convert $1 \frac{3}{10}$ to an improper fraction.

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Step 2.2.2.1

A mixed number is an addition of its whole and fractional parts.

$y^{1 + \frac{3}{10}} = p$

Step 2.2.2.2

Add $1$ and $\frac{3}{10}$ .

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Step 2.2.2.2.1

Write $1$ as a fraction with a common denominator.

$y^{\frac{10}{10} + \frac{3}{10}} = p$

Step 2.2.2.2.2

Combine the numerators over the common denominator.

$y^{\frac{10 + 3}{10}} = p$

Step 2.2.2.2.3

Add $10$ and $3$ .

$y^{\frac{13}{10}} = p$

Step 2.3

Raise each side of the equation to the power of $\frac{1}{1.3}$ to eliminate the fractional exponent on the left side.

${(y^{\frac{13}{10}})}^{\frac{1}{1.3}} = p^{\frac{1}{1.3}}$

Step 2.4

Simplify the exponent.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Simplify ${(y^{\frac{13}{10}})}^{\frac{1}{1.3}}$ .

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Step 2.4.1.1.1

Multiply the exponents in ${(y^{\frac{13}{10}})}^{\frac{1}{1.3}}$ .

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Step 2.4.1.1.1.1

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

$y^{\frac{13}{10} \cdot \frac{1}{1.3}} = p^{\frac{1}{1.3}}$

Step 2.4.1.1.1.2

Cancel the common factor of $1.3$ .

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Step 2.4.1.1.1.2.1

Factor $1.3$ out of $13$ .

$y^{\frac{1.3 (10)}{10} \cdot \frac{1}{1.3}} = p^{\frac{1}{1.3}}$

Step 2.4.1.1.1.2.2

Cancel the common factor.

$y^{\frac{1.3 \cdot 10}{10} \cdot \frac{1}{1.3}} = p^{\frac{1}{1.3}}$

Step 2.4.1.1.1.2.3

Rewrite the expression.

$y^{\frac{10}{10}} = p^{\frac{1}{1.3}}$

Step 2.4.1.1.1.3

Divide $10$ by $10$ .

$y^{1} = p^{\frac{1}{1.3}}$

Step 2.4.1.1.2

Simplify.

$y = p^{\frac{1}{1.3}}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Divide $1$ by $1.3$ .

$y = p^{0.\overline{769230}}$

Step 3

Replace $y$ with $f^{- 1} (p)$ to show the final answer.

$f^{- 1} (p) = p^{0.\overline{769230}}$

Step 4

Verify if $f^{- 1} (p) = p^{0.\overline{769230}}$ is the inverse of $f (p) = p^{1.3}$ .

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Step 4.1

To verify the inverse, check if $f^{- 1} (f (p)) = p$ and $f (f^{- 1} (p)) = p$ .

Step 4.2

Evaluate $f^{- 1} (f (p))$ .

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Step 4.2.1

Set up the composite result function.

$f^{- 1} (f (p))$

Step 4.2.2

Evaluate $f^{- 1} (p^{1.3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (p^{1.3}) = {(p^{1.3})}^{0.\overline{769230}}$

Step 4.2.3

Multiply the exponents in ${(p^{1.3})}^{0.\overline{769230}}$ .

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Step 4.2.3.1

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

$f^{- 1} (p^{1.3}) = p^{1.3 \cdot 0.\overline{769230}}$

Step 4.2.3.2

Multiply $1.3$ by $0.\overline{769230}$ .

$f^{- 1} (p^{1.3}) = p$

Step 4.3

Evaluate $f (f^{- 1} (p))$ .

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Step 4.3.1

Set up the composite result function.

$f (f^{- 1} (p))$

Step 4.3.2

Evaluate $f (p^{0.\overline{769230}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (p^{0.\overline{769230}}) = {(p^{0.\overline{769230}})}^{1.3}$

Step 4.3.3

Multiply the exponents in ${(p^{0.\overline{769230}})}^{1.3}$ .

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Step 4.3.3.1

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

$f (p^{0.\overline{769230}}) = p^{0.\overline{769230} \cdot 1.3}$

Step 4.3.3.2

Multiply $0.\overline{769230}$ by $1.3$ .

$f (p^{0.\overline{769230}}) = p$

Step 4.4

Since $f^{- 1} (f (p)) = p$ and $f (f^{- 1} (p)) = p$ , then $f^{- 1} (p) = p^{0.\overline{769230}}$ is the inverse of $f (p) = p^{1.3}$ .

$f^{- 1} (p) = p^{0.\overline{769230}}$