Pre-Algebra Examples

$\sqrt{x} - 5 \sqrt[4]{x} - 24 = 0$

Step 1

Factor $\sqrt{x} - 5 \sqrt[4]{x} - 24$ .

Tap for more steps...

Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$x^{\frac{1}{2}} - 5 \sqrt[4]{x} - 24 = 0$

Step 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$x^{\frac{1}{2}} - 5 x^{\frac{1}{4}} - 24 = 0$

Step 1.3

Rewrite $x^{\frac{1}{2}}$ as ${(x^{\frac{1}{4}})}^{2}$ .

${(x^{\frac{1}{4}})}^{2} - 5 x^{\frac{1}{4}} - 24 = 0$

Step 1.4

Let $u = x^{\frac{1}{4}}$ . Substitute $u$ for all occurrences of $x^{\frac{1}{4}}$ .

$u^{2} - 5 u - 24 = 0$

Step 1.5

Factor $u^{2} - 5 u - 24$ using the AC method.

Tap for more steps...

Step 1.5.1

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 $- 24$ and whose sum is $- 5$ .

$- 8, 3$

Step 1.5.2

Write the factored form using these integers.

$(u - 8) (u + 3) = 0$

Step 1.6

Replace all occurrences of $u$ with $x^{\frac{1}{4}}$ .

$(x^{\frac{1}{4}} - 8) (x^{\frac{1}{4}} + 3) = 0$

Step 2

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

$x^{\frac{1}{4}} - 8 = 0$

$x^{\frac{1}{4}} + 3 = 0$

Step 3

Set $x^{\frac{1}{4}} - 8$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.1

Set $x^{\frac{1}{4}} - 8$ equal to $0$ .

$x^{\frac{1}{4}} - 8 = 0$

Step 3.2

Solve $x^{\frac{1}{4}} - 8 = 0$ for $x$ .

Tap for more steps...

Step 3.2.1

Add $8$ to both sides of the equation.

$x^{\frac{1}{4}} = 8$

Step 3.2.2

Raise each side of the equation to the power of $4$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{4}})}^{4} = 8^{4}$

Step 3.2.3

Simplify the exponent.

Tap for more steps...

Step 3.2.3.1

Simplify the left side.

Tap for more steps...

Step 3.2.3.1.1

Simplify ${(x^{\frac{1}{4}})}^{4}$ .

Tap for more steps...

Step 3.2.3.1.1.1

Multiply the exponents in ${(x^{\frac{1}{4}})}^{4}$ .

Tap for more steps...

Step 3.2.3.1.1.1.1

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

$x^{\frac{1}{4} \cdot 4} = 8^{4}$

Step 3.2.3.1.1.1.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.2.3.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{4} \cdot 4} = 8^{4}$

Step 3.2.3.1.1.1.2.2

Rewrite the expression.

$x^{1} = 8^{4}$

Step 3.2.3.1.1.2

Simplify.

$x = 8^{4}$

Step 3.2.3.2

Simplify the right side.

Tap for more steps...

Step 3.2.3.2.1

Raise $8$ to the power of $4$ .

$x = 4096$

Step 4

Set $x^{\frac{1}{4}} + 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.1

Set $x^{\frac{1}{4}} + 3$ equal to $0$ .

$x^{\frac{1}{4}} + 3 = 0$

Step 4.2

Solve $x^{\frac{1}{4}} + 3 = 0$ for $x$ .

Tap for more steps...

Step 4.2.1

Subtract $3$ from both sides of the equation.

$x^{\frac{1}{4}} = - 3$

Step 4.2.2

Raise each side of the equation to the power of $4$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{4}})}^{4} = {(- 3)}^{4}$

Step 4.2.3

Simplify the exponent.

Tap for more steps...

Step 4.2.3.1

Simplify the left side.

Tap for more steps...

Step 4.2.3.1.1

Simplify ${(x^{\frac{1}{4}})}^{4}$ .

Tap for more steps...

Step 4.2.3.1.1.1

Multiply the exponents in ${(x^{\frac{1}{4}})}^{4}$ .

Tap for more steps...

Step 4.2.3.1.1.1.1

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

$x^{\frac{1}{4} \cdot 4} = {(- 3)}^{4}$

Step 4.2.3.1.1.1.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.2.3.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{4} \cdot 4} = {(- 3)}^{4}$

Step 4.2.3.1.1.1.2.2

Rewrite the expression.

$x^{1} = {(- 3)}^{4}$

Step 4.2.3.1.1.2

Simplify.

$x = {(- 3)}^{4}$

Step 4.2.3.2

Simplify the right side.

Tap for more steps...

Step 4.2.3.2.1

Raise $- 3$ to the power of $4$ .

$x = 81$

Step 5

The final solution is all the values that make $(x^{\frac{1}{4}} - 8) (x^{\frac{1}{4}} + 3) = 0$ true.

$x = 4096, 81$

Step 6

Exclude the solutions that do not make $\sqrt{x} - 5 \sqrt[4]{x} - 24 = 0$ true.

$x = 4096$