Solve by Factoring x^(2/5)+x^(1/5)-6=0

$x^{\frac{2}{5}} + x^{\frac{1}{5}} - 6 = 0$

Step 1

Rewrite $x^{\frac{2}{5}}$ as ${(x^{\frac{1}{5}})}^{2}$ .

${(x^{\frac{1}{5}})}^{2} + x^{\frac{1}{5}} - 6 = 0$

Step 2

Let $u = x^{\frac{1}{5}}$ . Substitute $u$ for all occurrences of $x^{\frac{1}{5}}$ .

$u^{2} + u - 6 = 0$

Step 3

Factor $u^{2} + u - 6$ 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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Write the factored form using these integers.

$(u - 2) (u + 3) = 0$

Step 4

Replace all occurrences of $u$ with $x^{\frac{1}{5}}$ .

$(x^{\frac{1}{5}} - 2) (x^{\frac{1}{5}} + 3) = 0$

Step 5

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

$x^{\frac{1}{5}} - 2 = 0$

$x^{\frac{1}{5}} + 3 = 0$

Step 6

Set $x^{\frac{1}{5}} - 2$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{5}} - 2$ equal to $0$ .

$x^{\frac{1}{5}} - 2 = 0$

Solve $x^{\frac{1}{5}} - 2 = 0$ for $x$ .

Add $2$ to both sides of the equation.

$x^{\frac{1}{5}} = 2$

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{5}})}^{5} = 2^{5}$

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{1}{5}})}^{5}$ .

Multiply the exponents in ${(x^{\frac{1}{5}})}^{5}$ .

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

$x^{\frac{1}{5} \cdot 5} = 2^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$x^{\frac{1}{5} \cdot 5} = 2^{5}$

Rewrite the expression.

$x^{1} = 2^{5}$

Simplify.

$x = 2^{5}$

Simplify the right side.

Raise $2$ to the power of $5$ .

$x = 32$

Step 7

Set $x^{\frac{1}{5}} + 3$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{5}} + 3$ equal to $0$ .

$x^{\frac{1}{5}} + 3 = 0$

Solve $x^{\frac{1}{5}} + 3 = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$x^{\frac{1}{5}} = - 3$

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{5}})}^{5} = {(- 3)}^{5}$

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{1}{5}})}^{5}$ .

Multiply the exponents in ${(x^{\frac{1}{5}})}^{5}$ .

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

$x^{\frac{1}{5} \cdot 5} = {(- 3)}^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$x^{\frac{1}{5} \cdot 5} = {(- 3)}^{5}$

Rewrite the expression.

$x^{1} = {(- 3)}^{5}$

Simplify.

$x = {(- 3)}^{5}$

Simplify the right side.

Raise $- 3$ to the power of $5$ .

$x = - 243$

Step 8

The final solution is all the values that make $(x^{\frac{1}{5}} - 2) (x^{\frac{1}{5}} + 3) = 0$ true.

$x = 32, - 243$

Solve by Factoring x^(2/5)+x^(1/5)-6=0

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