Algebra Examples

$\sqrt{p} - 2 \sqrt[4]{p} - 3 = 0$

Step 1

Factor $\sqrt{p} - 2 \sqrt[4]{p} - 3$ .

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Step 1.1

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

$p^{\frac{1}{2}} - 2 \sqrt[4]{p} - 3 = 0$

Step 1.2

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

$p^{\frac{1}{2}} - 2 p^{\frac{1}{4}} - 3 = 0$

Step 1.3

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

${(p^{\frac{1}{4}})}^{2} - 2 p^{\frac{1}{4}} - 3 = 0$

Step 1.4

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

$u^{2} - 2 u - 3 = 0$

Step 1.5

Factor $u^{2} - 2 u - 3$ using the AC method.

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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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 1.5.2

Write the factored form using these integers.

$(u - 3) (u + 1) = 0$

Step 1.6

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

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

Step 2

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

$p^{\frac{1}{4}} - 3 = 0$

$p^{\frac{1}{4}} + 1 = 0$

Step 3

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

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Step 3.1

Set $p^{\frac{1}{4}} - 3$ equal to $0$ .

$p^{\frac{1}{4}} - 3 = 0$

Step 3.2

Solve $p^{\frac{1}{4}} - 3 = 0$ for $p$ .

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Step 3.2.1

Add $3$ to both sides of the equation.

$p^{\frac{1}{4}} = 3$

Step 3.2.2

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

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

Step 3.2.3

Simplify the exponent.

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Step 3.2.3.1

Simplify the left side.

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Step 3.2.3.1.1

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

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Step 3.2.3.1.1.1

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

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Step 3.2.3.1.1.1.1

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

$p^{\frac{1}{4} \cdot 4} = 3^{4}$

Step 3.2.3.1.1.1.2

Cancel the common factor of $4$ .

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Step 3.2.3.1.1.1.2.1

Cancel the common factor.

$p^{\frac{1}{4} \cdot 4} = 3^{4}$

Step 3.2.3.1.1.1.2.2

Rewrite the expression.

$p^{1} = 3^{4}$

Step 3.2.3.1.1.2

Simplify.

$p = 3^{4}$

Step 3.2.3.2

Simplify the right side.

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Step 3.2.3.2.1

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

$p = 81$

Step 4

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

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Step 4.1

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

$p^{\frac{1}{4}} + 1 = 0$

Step 4.2

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

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$p^{\frac{1}{4}} = - 1$

Step 4.2.2

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

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

Step 4.2.3

Simplify the exponent.

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Step 4.2.3.1

Simplify the left side.

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Step 4.2.3.1.1

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

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Step 4.2.3.1.1.1

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

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Step 4.2.3.1.1.1.1

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

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

Step 4.2.3.1.1.1.2

Cancel the common factor of $4$ .

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Step 4.2.3.1.1.1.2.1

Cancel the common factor.

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

Step 4.2.3.1.1.1.2.2

Rewrite the expression.

$p^{1} = {(- 1)}^{4}$

Step 4.2.3.1.1.2

Simplify.

$p = {(- 1)}^{4}$

Step 4.2.3.2

Simplify the right side.

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Step 4.2.3.2.1

Raise $- 1$ to the power of $4$ .

$p = 1$

Step 5

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

$p = 81, 1$

Step 6

Exclude the solutions that do not make $\sqrt{p} - 2 \sqrt[4]{p} - 3 = 0$ true.

$p = 81$