Find All Complex Solutions x^(4/3)-10x^(2/3)+9=0

$x^{\frac{4}{3}} - 10 x^{\frac{2}{3}} + 9 = 0$

Step 1

Factor the left side of the equation.

Rewrite $x^{\frac{4}{3}}$ as ${(x^{\frac{2}{3}})}^{2}$ .

${(x^{\frac{2}{3}})}^{2} - 10 x^{\frac{2}{3}} + 9 = 0$

Let $u = x^{\frac{2}{3}}$ . Substitute $u$ for all occurrences of $x^{\frac{2}{3}}$ .

$u^{2} - 10 u + 9 = 0$

Factor $u^{2} - 10 u + 9$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $9$ and whose sum is $- 10$ .

$- 9, - 1$

Write the factored form using these integers.

$(u - 9) (u - 1) = 0$

Replace all occurrences of $u$ with $x^{\frac{2}{3}}$ .

$(x^{\frac{2}{3}} - 9) (x^{\frac{2}{3}} - 1) = 0$

Rewrite $x^{\frac{2}{3}}$ as ${(x^{\frac{1}{3}})}^{2}$ .

$({(x^{\frac{1}{3}})}^{2} - 9) (x^{\frac{2}{3}} - 1) = 0$

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

$({(x^{\frac{1}{3}})}^{2} - 3^{2}) (x^{\frac{2}{3}} - 1) = 0$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{\frac{1}{3}}$ and $b = 3$ .

$(x^{\frac{1}{3}} + 3) (x^{\frac{1}{3}} - 3) (x^{\frac{2}{3}} - 1) = 0$

Rewrite $x^{\frac{2}{3}}$ as ${(x^{\frac{1}{3}})}^{2}$ .

$(x^{\frac{1}{3}} + 3) (x^{\frac{1}{3}} - 3) ({(x^{\frac{1}{3}})}^{2} - 1) = 0$

Rewrite $1$ as $1^{2}$ .

$(x^{\frac{1}{3}} + 3) (x^{\frac{1}{3}} - 3) ({(x^{\frac{1}{3}})}^{2} - 1^{2}) = 0$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{\frac{1}{3}}$ and $b = 1$ .

$(x^{\frac{1}{3}} + 3) (x^{\frac{1}{3}} - 3) ((x^{\frac{1}{3}} + 1) (x^{\frac{1}{3}} - 1)) = 0$

Remove unnecessary parentheses.

$(x^{\frac{1}{3}} + 3) (x^{\frac{1}{3}} - 3) (x^{\frac{1}{3}} + 1) (x^{\frac{1}{3}} - 1) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{\frac{1}{3}} + 3 = 0$

$x^{\frac{1}{3}} - 3 = 0$

$x^{\frac{1}{3}} + 1 = 0$

$x^{\frac{1}{3}} - 1 = 0$

Step 3

Set $x^{\frac{1}{3}} + 3$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{3}} + 3$ equal to $0$ .

$x^{\frac{1}{3}} + 3 = 0$

Solve $x^{\frac{1}{3}} + 3 = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$x^{\frac{1}{3}} = - 3$

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{3}})}^{3} = {(- 3)}^{3}$

Simplify the exponent.

Simplify the left side.

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

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

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

$x^{\frac{1}{3} \cdot 3} = {(- 3)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = {(- 3)}^{3}$

Rewrite the expression.

$x^{1} = {(- 3)}^{3}$

Simplify.

$x = {(- 3)}^{3}$

Simplify the right side.

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

$x = - 27$

Step 4

Set $x^{\frac{1}{3}} - 3$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{3}} - 3$ equal to $0$ .

$x^{\frac{1}{3}} - 3 = 0$

Solve $x^{\frac{1}{3}} - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$x^{\frac{1}{3}} = 3$

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

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

Simplify the exponent.

Simplify the left side.

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

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

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

$x^{\frac{1}{3} \cdot 3} = 3^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = 3^{3}$

Rewrite the expression.

$x^{1} = 3^{3}$

Simplify.

$x = 3^{3}$

Simplify the right side.

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

$x = 27$

Step 5

Set $x^{\frac{1}{3}} + 1$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{3}} + 1$ equal to $0$ .

$x^{\frac{1}{3}} + 1 = 0$

Solve $x^{\frac{1}{3}} + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$x^{\frac{1}{3}} = - 1$

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Simplify the exponent.

Simplify the left side.

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

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

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

$x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Rewrite the expression.

$x^{1} = {(- 1)}^{3}$

Simplify.

$x = {(- 1)}^{3}$

Simplify the right side.

Raise $- 1$ to the power of $3$ .

$x = - 1$

Step 6

Set $x^{\frac{1}{3}} - 1$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{3}} - 1$ equal to $0$ .

$x^{\frac{1}{3}} - 1 = 0$

Solve $x^{\frac{1}{3}} - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$x^{\frac{1}{3}} = 1$

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

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

Simplify the exponent.

Simplify the left side.

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

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

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

$x^{\frac{1}{3} \cdot 3} = 1^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = 1^{3}$

Rewrite the expression.

$x^{1} = 1^{3}$

Simplify.

$x = 1^{3}$

Simplify the right side.

One to any power is one.

$x = 1$

Step 7

The final solution is all the values that make $(x^{\frac{1}{3}} + 3) (x^{\frac{1}{3}} - 3) (x^{\frac{1}{3}} + 1) (x^{\frac{1}{3}} - 1) = 0$ true.

$x = - 27, 27, - 1, 1$

Find All Complex Solutions x^(4/3)-10x^(2/3)+9=0

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