Find the Inverse t(d)=(12.5/d)^3

$t (d) = {(\frac{12.5}{d})}^{3}$

Step 1

Write $t (d) = {(\frac{12.5}{d})}^{3}$ as an equation.

$y = {(\frac{12.5}{d})}^{3}$

Step 2

Interchange the variables.

$d = {(\frac{12.5}{y})}^{3}$

Step 3

Solve for $y$ .

Rewrite the equation as ${(\frac{12.5}{y})}^{3} = d$ .

${(\frac{12.5}{y})}^{3} = d$

Take the cube root of both sides of the equation to eliminate the exponent on the left side.

$\frac{12.5}{y} = \sqrt[3]{d}$

Find the LCD of the terms in the equation.

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

$y, 1$

The LCM of one and any expression is the expression.

$y$

Multiply each term in $\frac{12.5}{y} = \sqrt[3]{d}$ by $y$ to eliminate the fractions.

Multiply each term in $\frac{12.5}{y} = \sqrt[3]{d}$ by $y$ .

$\frac{12.5}{y} y = \sqrt[3]{d} y$

Simplify the left side.

Cancel the common factor of $y$ .

Cancel the common factor.

$\frac{12.5}{y} y = \sqrt[3]{d} y$

Rewrite the expression.

$12.5 = \sqrt[3]{d} y$

Solve the equation.

Rewrite the equation as $\sqrt[3]{d} y = 12.5$ .

$\sqrt[3]{d} y = 12.5$

Divide each term in $\sqrt[3]{d} y = 12.5$ by $\sqrt[3]{d}$ and simplify.

Divide each term in $\sqrt[3]{d} y = 12.5$ by $\sqrt[3]{d}$ .

$\frac{\sqrt[3]{d} y}{\sqrt[3]{d}} = \frac{12.5}{\sqrt[3]{d}}$

Simplify the left side.

Cancel the common factor of $\sqrt[3]{d}$ .

Cancel the common factor.

$\frac{\sqrt[3]{d} y}{\sqrt[3]{d}} = \frac{12.5}{\sqrt[3]{d}}$

Divide $y$ by $1$ .

$y = \frac{12.5}{\sqrt[3]{d}}$

Simplify the right side.

Multiply $\frac{12.5}{\sqrt[3]{d}}$ by $\frac{{\sqrt[3]{d}}^{2}}{{\sqrt[3]{d}}^{2}}$ .

$y = \frac{12.5}{\sqrt[3]{d}} \cdot \frac{{\sqrt[3]{d}}^{2}}{{\sqrt[3]{d}}^{2}}$

Combine and simplify the denominator.

Multiply $\frac{12.5}{\sqrt[3]{d}}$ by $\frac{{\sqrt[3]{d}}^{2}}{{\sqrt[3]{d}}^{2}}$ .

$y = \frac{12.5 {\sqrt[3]{d}}^{2}}{\sqrt[3]{d} {\sqrt[3]{d}}^{2}}$

Raise $\sqrt[3]{d}$ to the power of $1$ .

$y = \frac{12.5 {\sqrt[3]{d}}^{2}}{{\sqrt[3]{d}}^{1} {\sqrt[3]{d}}^{2}}$

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

$y = \frac{12.5 {\sqrt[3]{d}}^{2}}{{\sqrt[3]{d}}^{1 + 2}}$

Add $1$ and $2$ .

$y = \frac{12.5 {\sqrt[3]{d}}^{2}}{{\sqrt[3]{d}}^{3}}$

Rewrite ${\sqrt[3]{d}}^{3}$ as $d$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{d}$ as $d^{\frac{1}{3}}$ .

$y = \frac{12.5 {\sqrt[3]{d}}^{2}}{{(d^{\frac{1}{3}})}^{3}}$

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

$y = \frac{12.5 {\sqrt[3]{d}}^{2}}{d^{\frac{1}{3} \cdot 3}}$

Combine $\frac{1}{3}$ and $3$ .

$y = \frac{12.5 {\sqrt[3]{d}}^{2}}{d^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$y = \frac{12.5 {\sqrt[3]{d}}^{2}}{d^{\frac{3}{3}}}$

Rewrite the expression.

$y = \frac{12.5 {\sqrt[3]{d}}^{2}}{d^{1}}$

Simplify.

$y = \frac{12.5 {\sqrt[3]{d}}^{2}}{d}$

Rewrite ${\sqrt[3]{d}}^{2}$ as ${(d^{2})}^{\frac{1}{3}}$ .

$y = \frac{12.5 \sqrt[3]{d^{2}}}{d}$

Step 4

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

$t^{- 1} (d) = \frac{12.5 \sqrt[3]{d^{2}}}{d}$

Step 5

Verify if $t^{- 1} (d) = \frac{12.5 \sqrt[3]{d^{2}}}{d}$ is the inverse of $t (d) = {(\frac{12.5}{d})}^{3}$ .

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

Evaluate $t^{- 1} (t (d))$ .

Set up the composite result function.

$t^{- 1} (t (d))$

Evaluate $t^{- 1} ({(\frac{12.5}{d})}^{3})$ by substituting in the value of $t$ into $t^{- 1}$ .

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{12.5 \sqrt[3]{{({(\frac{12.5}{d})}^{3})}^{2}}}{{(\frac{12.5}{d})}^{3}}$

Simplify the numerator.

Apply the product rule to $\frac{12.5}{d}$ .

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{12.5 \sqrt[3]{{(\frac{{12.5}^{3}}{d^{3}})}^{2}}}{{(\frac{12.5}{d})}^{3}}$

Apply the product rule to $\frac{{12.5}^{3}}{d^{3}}$ .

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{12.5 \sqrt[3]{\frac{{({12.5}^{3})}^{2}}{{(d^{3})}^{2}}}}{{(\frac{12.5}{d})}^{3}}$

Simplify the numerator.

Multiply the exponents in ${({12.5}^{3})}^{2}$ .

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

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{12.5 \sqrt[3]{\frac{{12.5}^{3 \cdot 2}}{{(d^{3})}^{2}}}}{{(\frac{12.5}{d})}^{3}}$

Multiply $3$ by $2$ .

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{12.5 \sqrt[3]{\frac{{12.5}^{6}}{{(d^{3})}^{2}}}}{{(\frac{12.5}{d})}^{3}}$

Raise $12.5$ to the power of $6$ .

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{12.5 \sqrt[3]{\frac{3814697.265625}{{(d^{3})}^{2}}}}{{(\frac{12.5}{d})}^{3}}$

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

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

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{12.5 \sqrt[3]{\frac{3814697.265625}{d^{3 \cdot 2}}}}{{(\frac{12.5}{d})}^{3}}$

Multiply $3$ by $2$ .

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{12.5 \sqrt[3]{\frac{3814697.265625}{d^{6}}}}{{(\frac{12.5}{d})}^{3}}$

Rewrite $\frac{3814697.265625}{d^{6}}$ as ${(\frac{1}{d^{2}})}^{3} \cdot 3814697.265625$ .

Factor the perfect power $1^{3}$ out of $3814697.265625$ .

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{12.5 \sqrt[3]{\frac{1^{3} \cdot 3814697.265625}{d^{6}}}}{{(\frac{12.5}{d})}^{3}}$

Factor the perfect power ${(d^{2})}^{3}$ out of $d^{6}$ .

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{12.5 \sqrt[3]{\frac{1^{3} \cdot 3814697.265625}{{(d^{2})}^{3} \cdot 1}}}{{(\frac{12.5}{d})}^{3}}$

Rearrange the fraction $\frac{1^{3} \cdot 3814697.265625}{{(d^{2})}^{3} \cdot 1}$ .

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{12.5 \sqrt[3]{{(\frac{1}{d^{2}})}^{3} \cdot 3814697.265625}}{{(\frac{12.5}{d})}^{3}}$

Pull terms out from under the radical.

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{12.5 (\frac{1}{d^{2}} \cdot \sqrt[3]{3814697.265625})}{{(\frac{12.5}{d})}^{3}}$

Combine $\frac{1}{d^{2}}$ and $\sqrt[3]{3814697.265625}$ .

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{12.5 (\frac{\sqrt[3]{3814697.265625}}{d^{2}})}{{(\frac{12.5}{d})}^{3}}$

Simplify the denominator.

Apply the product rule to $\frac{12.5}{d}$ .

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{12.5 (\frac{\sqrt[3]{3814697.265625}}{d^{2}})}{\frac{{12.5}^{3}}{d^{3}}}$

Raise $12.5$ to the power of $3$ .

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{12.5 (\frac{\sqrt[3]{3814697.265625}}{d^{2}})}{\frac{1953.125}{d^{3}}}$

Simplify terms.

Combine $12.5$ and $\frac{\sqrt[3]{3814697.265625}}{d^{2}}$ .

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{\frac{12.5 \sqrt[3]{3814697.265625}}{d^{2}}}{\frac{1953.125}{d^{3}}}$

Multiply $12.5$ by $\sqrt[3]{3814697.265625}$ .

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{\frac{1953.125}{d^{2}}}{\frac{1953.125}{d^{3}}}$

Multiply the numerator by the reciprocal of the denominator.

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{1953.125}{d^{2}} \cdot \frac{d^{3}}{1953.125}$

Combine.

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{1953.125 d^{3}}{d^{2} \cdot 1953.125}$

Cancel the common factor of $1953.125$ .

Cancel the common factor.

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{1953.125 d^{3}}{d^{2} \cdot 1953.125}$

Rewrite the expression.

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{d^{3}}{d^{2}}$

Cancel the common factor of $d^{3}$ and $d^{2}$ .

Factor $d^{2}$ out of $d^{3}$ .

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{d^{2} d}{d^{2}}$

Cancel the common factors.

Multiply by $1$ .

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{d^{2} d}{d^{2} \cdot 1}$

Cancel the common factor.

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{d^{2} d}{d^{2} \cdot 1}$

Rewrite the expression.

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = \frac{d}{1}$

Divide $d$ by $1$ .

$t^{- 1} ({(\frac{12.5}{d})}^{3}) = d$

Evaluate $t (t^{- 1} (d))$ .

Set up the composite result function.

$t (t^{- 1} (d))$

Evaluate $t (\frac{12.5 \sqrt[3]{d^{2}}}{d})$ by substituting in the value of $t^{- 1}$ into $t$ .

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{12.5}{\frac{12.5 \sqrt[3]{d^{2}}}{d}})}^{3}$

Multiply the numerator by the reciprocal of the denominator.

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(12.5 (\frac{d}{12.5 \sqrt[3]{d^{2}}}))}^{3}$

Cancel the common factor of $12.5$ .

Factor $12.5$ out of $12.5 \sqrt[3]{d^{2}}$ .

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(12.5 (\frac{d}{12.5 (\sqrt[3]{d^{2}})}))}^{3}$

Cancel the common factor.

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(12.5 (\frac{d}{12.5 \sqrt[3]{d^{2}}}))}^{3}$

Rewrite the expression.

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{d}{\sqrt[3]{d^{2}}})}^{3}$

Multiply $\frac{d}{\sqrt[3]{d^{2}}}$ by $\frac{{\sqrt[3]{d^{2}}}^{2}}{{\sqrt[3]{d^{2}}}^{2}}$ .

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{d}{\sqrt[3]{d^{2}}} \cdot \frac{{\sqrt[3]{d^{2}}}^{2}}{{\sqrt[3]{d^{2}}}^{2}})}^{3}$

Simplify terms.

Combine and simplify the denominator.

Multiply $\frac{d}{\sqrt[3]{d^{2}}}$ by $\frac{{\sqrt[3]{d^{2}}}^{2}}{{\sqrt[3]{d^{2}}}^{2}}$ .

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{d {\sqrt[3]{d^{2}}}^{2}}{\sqrt[3]{d^{2}} {\sqrt[3]{d^{2}}}^{2}})}^{3}$

Raise $\sqrt[3]{d^{2}}$ to the power of $1$ .

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{d {\sqrt[3]{d^{2}}}^{2}}{\sqrt[3]{d^{2}} {\sqrt[3]{d^{2}}}^{2}})}^{3}$

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

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{d {\sqrt[3]{d^{2}}}^{2}}{{\sqrt[3]{d^{2}}}^{1 + 2}})}^{3}$

Add $1$ and $2$ .

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{d {\sqrt[3]{d^{2}}}^{2}}{{\sqrt[3]{d^{2}}}^{3}})}^{3}$

Rewrite ${\sqrt[3]{d^{2}}}^{3}$ as $d^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{d^{2}}$ as $d^{\frac{2}{3}}$ .

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{d {\sqrt[3]{d^{2}}}^{2}}{{(d^{\frac{2}{3}})}^{3}})}^{3}$

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

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{d {\sqrt[3]{d^{2}}}^{2}}{d^{\frac{2}{3} \cdot 3}})}^{3}$

Combine $\frac{2}{3}$ and $3$ .

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{d {\sqrt[3]{d^{2}}}^{2}}{d^{\frac{2 \cdot 3}{3}}})}^{3}$

Multiply $2$ by $3$ .

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{d {\sqrt[3]{d^{2}}}^{2}}{d^{\frac{6}{3}}})}^{3}$

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6$ .

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{d {\sqrt[3]{d^{2}}}^{2}}{d^{\frac{3 \cdot 2}{3}}})}^{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{d {\sqrt[3]{d^{2}}}^{2}}{d^{\frac{3 \cdot 2}{3 (1)}}})}^{3}$

Cancel the common factor.

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{d {\sqrt[3]{d^{2}}}^{2}}{d^{\frac{3 \cdot 2}{3 \cdot 1}}})}^{3}$

Rewrite the expression.

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{d {\sqrt[3]{d^{2}}}^{2}}{d^{\frac{2}{1}}})}^{3}$

Divide $2$ by $1$ .

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{d {\sqrt[3]{d^{2}}}^{2}}{d^{2}})}^{3}$

Cancel the common factor of $d$ and $d^{2}$ .

Factor $d$ out of $d {\sqrt[3]{d^{2}}}^{2}$ .

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{d ({\sqrt[3]{d^{2}}}^{2})}{d^{2}})}^{3}$

Cancel the common factors.

Factor $d$ out of $d^{2}$ .

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{d ({\sqrt[3]{d^{2}}}^{2})}{d \cdot d})}^{3}$

Cancel the common factor.

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{d {\sqrt[3]{d^{2}}}^{2}}{d \cdot d})}^{3}$

Rewrite the expression.

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{{\sqrt[3]{d^{2}}}^{2}}{d})}^{3}$

Simplify the numerator.

Rewrite ${\sqrt[3]{d^{2}}}^{2}$ as ${({(d^{2})}^{2})}^{\frac{1}{3}}$ .

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{\sqrt[3]{{(d^{2})}^{2}}}{d})}^{3}$

Multiply the exponents in ${(d^{2})}^{2}$ .

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

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{\sqrt[3]{d^{2 \cdot 2}}}{d})}^{3}$

Multiply $2$ by $2$ .

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{\sqrt[3]{d^{4}}}{d})}^{3}$

Factor out $d^{3}$ .

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{\sqrt[3]{d^{3} d}}{d})}^{3}$

Pull terms out from under the radical.

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{d \sqrt[3]{d}}{d})}^{3}$

Simplify terms.

Cancel the common factor of $d$ .

Cancel the common factor.

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(\frac{d \sqrt[3]{d}}{d})}^{3}$

Divide $\sqrt[3]{d}$ by $1$ .

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {\sqrt[3]{d}}^{3}$

Rewrite ${\sqrt[3]{d}}^{3}$ as $d$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{d}$ as $d^{\frac{1}{3}}$ .

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = {(d^{\frac{1}{3}})}^{3}$

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

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = d^{\frac{1}{3} \cdot 3}$

Combine $\frac{1}{3}$ and $3$ .

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = d^{\frac{3}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = d^{\frac{3}{3}}$

Rewrite the expression.

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = d$

Simplify.

$t (\frac{12.5 \sqrt[3]{d^{2}}}{d}) = d$

Since $t^{- 1} (t (d)) = d$ and $t (t^{- 1} (d)) = d$ , then $t^{- 1} (d) = \frac{12.5 \sqrt[3]{d^{2}}}{d}$ is the inverse of $t (d) = {(\frac{12.5}{d})}^{3}$ .

$t^{- 1} (d) = \frac{12.5 \sqrt[3]{d^{2}}}{d}$

Find the Inverse t(d)=(12.5/d)^3

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