Find the Inverse cube root of x-7

$\sqrt[3]{x} - 7$

Step 1

Interchange the variables.

$x = \sqrt[3]{y} - 7$

Step 2

Solve for $y$ .

Rewrite the equation as $\sqrt[3]{y} - 7 = x$ .

$\sqrt[3]{y} - 7 = x$

Add $7$ to both sides of the equation.

$\sqrt[3]{y} = x + 7$

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{y}}^{3} = {(x + 7)}^{3}$

Simplify each side of the equation.

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

${(y^{\frac{1}{3}})}^{3} = {(x + 7)}^{3}$

Simplify the left side.

Simplify ${(y^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${(y^{\frac{1}{3}})}^{3}$ .

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

$y^{\frac{1}{3} \cdot 3} = {(x + 7)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(x + 7)}^{3}$

Rewrite the expression.

$y^{1} = {(x + 7)}^{3}$

Simplify.

$y = {(x + 7)}^{3}$

Simplify the right side.

Simplify ${(x + 7)}^{3}$ .

Use the Binomial Theorem.

$y = x^{3} + 3 x^{2} \cdot 7 + 3 x \cdot 7^{2} + 7^{3}$

Simplify each term.

Multiply $7$ by $3$ .

$y = x^{3} + 21 x^{2} + 3 x \cdot 7^{2} + 7^{3}$

Raise $7$ to the power of $2$ .

$y = x^{3} + 21 x^{2} + 3 x \cdot 49 + 7^{3}$

Multiply $49$ by $3$ .

$y = x^{3} + 21 x^{2} + 147 x + 7^{3}$

Raise $7$ to the power of $3$ .

$y = x^{3} + 21 x^{2} + 147 x + 343$

Step 3

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

$f^{- 1} (x) = x^{3} + 21 x^{2} + 147 x + 343$

Step 4

Verify if $f^{- 1} (x) = x^{3} + 21 x^{2} + 147 x + 343$ is the inverse of $f (x) = \sqrt[3]{x} - 7$ .

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

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

Set up the composite result function.

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

Evaluate $f^{- 1} (\sqrt[3]{x} - 7)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt[3]{x} - 7) = {(\sqrt[3]{x} - 7)}^{3} + 21 {(\sqrt[3]{x} - 7)}^{2} + 147 (\sqrt[3]{x} - 7) + 343$

Simplify each term.

Use the Binomial Theorem.

$f^{- 1} (\sqrt[3]{x} - 7) = {\sqrt[3]{x}}^{3} + 3 {\sqrt[3]{x}}^{2} \cdot - 7 + 3 \sqrt[3]{x} {(- 7)}^{2} + {(- 7)}^{3} + 21 {(\sqrt[3]{x} - 7)}^{2} + 147 (\sqrt[3]{x} - 7) + 343$

Simplify each term.

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

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

$f^{- 1} (\sqrt[3]{x} - 7) = {(x^{\frac{1}{3}})}^{3} + 3 {\sqrt[3]{x}}^{2} \cdot - 7 + 3 \sqrt[3]{x} {(- 7)}^{2} + {(- 7)}^{3} + 21 {(\sqrt[3]{x} - 7)}^{2} + 147 (\sqrt[3]{x} - 7) + 343$

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

$f^{- 1} (\sqrt[3]{x} - 7) = x^{\frac{1}{3} \cdot 3} + 3 {\sqrt[3]{x}}^{2} \cdot - 7 + 3 \sqrt[3]{x} {(- 7)}^{2} + {(- 7)}^{3} + 21 {(\sqrt[3]{x} - 7)}^{2} + 147 (\sqrt[3]{x} - 7) + 343$

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

$f^{- 1} (\sqrt[3]{x} - 7) = x^{\frac{3}{3}} + 3 {\sqrt[3]{x}}^{2} \cdot - 7 + 3 \sqrt[3]{x} {(- 7)}^{2} + {(- 7)}^{3} + 21 {(\sqrt[3]{x} - 7)}^{2} + 147 (\sqrt[3]{x} - 7) + 343$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} (\sqrt[3]{x} - 7) = x^{\frac{3}{3}} + 3 {\sqrt[3]{x}}^{2} \cdot - 7 + 3 \sqrt[3]{x} {(- 7)}^{2} + {(- 7)}^{3} + 21 {(\sqrt[3]{x} - 7)}^{2} + 147 (\sqrt[3]{x} - 7) + 343$

Rewrite the expression.

$f^{- 1} (\sqrt[3]{x} - 7) = x + 3 {\sqrt[3]{x}}^{2} \cdot - 7 + 3 \sqrt[3]{x} {(- 7)}^{2} + {(- 7)}^{3} + 21 {(\sqrt[3]{x} - 7)}^{2} + 147 (\sqrt[3]{x} - 7) + 343$

Simplify.

$f^{- 1} (\sqrt[3]{x} - 7) = x + 3 {\sqrt[3]{x}}^{2} \cdot - 7 + 3 \sqrt[3]{x} {(- 7)}^{2} + {(- 7)}^{3} + 21 {(\sqrt[3]{x} - 7)}^{2} + 147 (\sqrt[3]{x} - 7) + 343$

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

$f^{- 1} (\sqrt[3]{x} - 7) = x + 3 \sqrt[3]{x^{2}} \cdot - 7 + 3 \sqrt[3]{x} {(- 7)}^{2} + {(- 7)}^{3} + 21 {(\sqrt[3]{x} - 7)}^{2} + 147 (\sqrt[3]{x} - 7) + 343$

Multiply $- 7$ by $3$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 3 \sqrt[3]{x} {(- 7)}^{2} + {(- 7)}^{3} + 21 {(\sqrt[3]{x} - 7)}^{2} + 147 (\sqrt[3]{x} - 7) + 343$

Raise $- 7$ to the power of $2$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 3 \sqrt[3]{x} \cdot 49 + {(- 7)}^{3} + 21 {(\sqrt[3]{x} - 7)}^{2} + 147 (\sqrt[3]{x} - 7) + 343$

Multiply $49$ by $3$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} + {(- 7)}^{3} + 21 {(\sqrt[3]{x} - 7)}^{2} + 147 (\sqrt[3]{x} - 7) + 343$

Raise $- 7$ to the power of $3$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 {(\sqrt[3]{x} - 7)}^{2} + 147 (\sqrt[3]{x} - 7) + 343$

Rewrite ${(\sqrt[3]{x} - 7)}^{2}$ as $(\sqrt[3]{x} - 7) (\sqrt[3]{x} - 7)$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 ((\sqrt[3]{x} - 7) (\sqrt[3]{x} - 7)) + 147 (\sqrt[3]{x} - 7) + 343$

Expand $(\sqrt[3]{x} - 7) (\sqrt[3]{x} - 7)$ using the FOIL Method.

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 (\sqrt[3]{x} (\sqrt[3]{x} - 7) - 7 (\sqrt[3]{x} - 7)) + 147 (\sqrt[3]{x} - 7) + 343$

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot - 7 - 7 (\sqrt[3]{x} - 7)) + 147 (\sqrt[3]{x} - 7) + 343$

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot - 7 - 7 \sqrt[3]{x} - 7 \cdot - 7) + 147 (\sqrt[3]{x} - 7) + 343$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt[3]{x} \sqrt[3]{x}$ .

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

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot - 7 - 7 \sqrt[3]{x} - 7 \cdot - 7) + 147 (\sqrt[3]{x} - 7) + 343$

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

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot - 7 - 7 \sqrt[3]{x} - 7 \cdot - 7) + 147 (\sqrt[3]{x} - 7) + 343$

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

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 ({\sqrt[3]{x}}^{1 + 1} + \sqrt[3]{x} \cdot - 7 - 7 \sqrt[3]{x} - 7 \cdot - 7) + 147 (\sqrt[3]{x} - 7) + 343$

Add $1$ and $1$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 ({\sqrt[3]{x}}^{2} + \sqrt[3]{x} \cdot - 7 - 7 \sqrt[3]{x} - 7 \cdot - 7) + 147 (\sqrt[3]{x} - 7) + 343$

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

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 (\sqrt[3]{x^{2}} + \sqrt[3]{x} \cdot - 7 - 7 \sqrt[3]{x} - 7 \cdot - 7) + 147 (\sqrt[3]{x} - 7) + 343$

Move $- 7$ to the left of $\sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 (\sqrt[3]{x^{2}} - 7 \cdot \sqrt[3]{x} - 7 \sqrt[3]{x} - 7 \cdot - 7) + 147 (\sqrt[3]{x} - 7) + 343$

Multiply $- 7$ by $- 7$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 (\sqrt[3]{x^{2}} - 7 \sqrt[3]{x} - 7 \sqrt[3]{x} + 49) + 147 (\sqrt[3]{x} - 7) + 343$

Subtract $7 \sqrt[3]{x}$ from $- 7 \sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 (\sqrt[3]{x^{2}} - 14 \sqrt[3]{x} + 49) + 147 (\sqrt[3]{x} - 7) + 343$

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 \sqrt[3]{x^{2}} + 21 (- 14 \sqrt[3]{x}) + 21 \cdot 49 + 147 (\sqrt[3]{x} - 7) + 343$

Simplify.

Multiply $- 14$ by $21$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} + 21 \cdot 49 + 147 (\sqrt[3]{x} - 7) + 343$

Multiply $21$ by $49$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} + 1029 + 147 (\sqrt[3]{x} - 7) + 343$

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} + 1029 + 147 \sqrt[3]{x} + 147 \cdot - 7 + 343$

Multiply $147$ by $- 7$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} + 1029 + 147 \sqrt[3]{x} - 1029 + 343$

Simplify by adding terms.

Combine the opposite terms in $x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} + 1029 + 147 \sqrt[3]{x} - 1029 + 343$ .

Add $- 21 \sqrt[3]{x^{2}}$ and $21 \sqrt[3]{x^{2}}$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x + 0 + 147 \sqrt[3]{x} - 343 - 294 \sqrt[3]{x} + 1029 + 147 \sqrt[3]{x} - 1029 + 343$

Add $x$ and $0$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x + 147 \sqrt[3]{x} - 343 - 294 \sqrt[3]{x} + 1029 + 147 \sqrt[3]{x} - 1029 + 343$

Subtract $1029$ from $1029$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x + 147 \sqrt[3]{x} - 343 - 294 \sqrt[3]{x} + 0 + 147 \sqrt[3]{x} + 343$

Add $x + 147 \sqrt[3]{x} - 343 - 294 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x + 147 \sqrt[3]{x} - 343 - 294 \sqrt[3]{x} + 147 \sqrt[3]{x} + 343$

Add $- 343$ and $343$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x + 147 \sqrt[3]{x} + 0 - 294 \sqrt[3]{x} + 147 \sqrt[3]{x}$

Add $x + 147 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x + 147 \sqrt[3]{x} - 294 \sqrt[3]{x} + 147 \sqrt[3]{x}$

Subtract $294 \sqrt[3]{x}$ from $147 \sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x - 147 \sqrt[3]{x} + 147 \sqrt[3]{x}$

Combine the opposite terms in $x - 147 \sqrt[3]{x} + 147 \sqrt[3]{x}$ .

Add $- 147 \sqrt[3]{x}$ and $147 \sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\sqrt[3]{x} - 7) = x$

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

Set up the composite result function.

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

Evaluate $f (x^{3} + 21 x^{2} + 147 x + 343)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{3} + 21 x^{2} + 147 x + 343) = \sqrt[3]{x^{3} + 21 x^{2} + 147 x + 343} - 7$

Remove parentheses.

$f (x^{3} + 21 x^{2} + 147 x + 343) = \sqrt[3]{x^{3} + 21 x^{2} + 147 x + 343} - 7$

Simplify each term.

Make each term match the terms from the binomial theorem formula.

$f (x^{3} + 21 x^{2} + 147 x + 343) = \sqrt[3]{x^{3} + 3 \cdot (7 x^{2}) + 3 \cdot (7^{2} x) + 7^{3}} - 7$

Factor $x^{3} + 3 \cdot 7 x^{2} + 3 \cdot 7^{2} x + 7^{3}$ using the binomial theorem.

$f (x^{3} + 21 x^{2} + 147 x + 343) = \sqrt[3]{{(x + 7)}^{3}} - 7$

Pull terms out from under the radical, assuming real numbers.

$f (x^{3} + 21 x^{2} + 147 x + 343) = x + 7 - 7$

Combine the opposite terms in $x + 7 - 7$ .

Subtract $7$ from $7$ .

$f (x^{3} + 21 x^{2} + 147 x + 343) = x + 0$

Add $x$ and $0$ .

$f (x^{3} + 21 x^{2} + 147 x + 343) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{3} + 21 x^{2} + 147 x + 343$ is the inverse of $f (x) = \sqrt[3]{x} - 7$ .

$f^{- 1} (x) = x^{3} + 21 x^{2} + 147 x + 343$

Find the Inverse cube root of x-7

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