Solve the Rational Equation for x x^(4/3)-10x^(2/3)+9=0

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

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 $x^{\frac{2}{3}}$ .

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

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

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

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

Set $x^{\frac{2}{3}} - 9$ equal to $0$ and solve for $x$ .

Set $x^{\frac{2}{3}} - 9$ equal to $0$ .

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

Solve $x^{\frac{2}{3}} - 9 = 0$ for $x$ .

Add $9$ to both sides of the equation.

$x^{\frac{2}{3}} = 9$

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 9^{\frac{3}{2}}$

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Multiply the exponents in ${(x^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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

$x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 9^{\frac{3}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 9^{\frac{3}{2}}$

Rewrite the expression.

$x^{\frac{1}{3} \cdot 3} = \pm 9^{\frac{3}{2}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = \pm 9^{\frac{3}{2}}$

Rewrite the expression.

$x^{1} = \pm 9^{\frac{3}{2}}$

Simplify.

$x = \pm 9^{\frac{3}{2}}$

Simplify the right side.

Simplify $\pm 9^{\frac{3}{2}}$ .

Simplify the expression.

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

$x = \pm {(3^{2})}^{\frac{3}{2}}$

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

$x = \pm 3^{2 (\frac{3}{2})}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm 3^{2 (\frac{3}{2})}$

Rewrite the expression.

$x = \pm 3^{3}$

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

$x = \pm 27$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 27$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 27$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 27, - 27$

Set $x^{\frac{2}{3}} - 1$ equal to $0$ and solve for $x$ .

Set $x^{\frac{2}{3}} - 1$ equal to $0$ .

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

Solve $x^{\frac{2}{3}} - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$x^{\frac{2}{3}} = 1$

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

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

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Multiply the exponents in ${(x^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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

$x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 1^{\frac{3}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 1^{\frac{3}{2}}$

Rewrite the expression.

$x^{\frac{1}{3} \cdot 3} = \pm 1^{\frac{3}{2}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = \pm 1^{\frac{3}{2}}$

Rewrite the expression.

$x^{1} = \pm 1^{\frac{3}{2}}$

Simplify.

$x = \pm 1^{\frac{3}{2}}$

Simplify the right side.

One to any power is one.

$x = \pm 1$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 1$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1, - 1$

The final solution is all the values that make $(x^{\frac{2}{3}} - 9) (x^{\frac{2}{3}} - 1) = 0$ true.

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

Solve the Rational Equation for x x^(4/3)-10x^(2/3)+9=0

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