Algebra Examples

$x^{\frac{7}{4}} + x^{\frac{5}{2}}$

Step 1

Factor $x^{\frac{7}{4}}$ out of $x^{\frac{7}{4}} + x^{\frac{5}{2}}$ .

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Step 1.1

Multiply by $1$ .

$x^{\frac{7}{4}} \cdot 1 + x^{\frac{5}{2}}$

Step 1.2

Factor $x^{\frac{7}{4}}$ out of $x^{\frac{5}{2}}$ .

$x^{\frac{7}{4}} \cdot 1 + x^{\frac{7}{4}} x^{\frac{3}{4}}$

Step 1.3

Factor $x^{\frac{7}{4}}$ out of $x^{\frac{7}{4}} \cdot 1 + x^{\frac{7}{4}} x^{\frac{3}{4}}$ .

$x^{\frac{7}{4}} (1 + x^{\frac{3}{4}})$

Step 2

Rewrite $1$ as $1^{3}$ .

$x^{\frac{7}{4}} (1^{3} + x^{\frac{3}{4}})$

Step 3

Rewrite $x^{\frac{3}{4}}$ as ${(x^{\frac{1}{4}})}^{3}$ .

$x^{\frac{7}{4}} (1^{3} + {(x^{\frac{1}{4}})}^{3})$

Step 4

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 1$ and $b = x^{\frac{1}{4}}$ .

$x^{\frac{7}{4}} ((1 + x^{\frac{1}{4}}) (1^{2} - 1 x^{\frac{1}{4}} + {(x^{\frac{1}{4}})}^{2}))$

Step 5

Factor.

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Step 5.1

Simplify.

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Step 5.1.1

One to any power is one.

$x^{\frac{7}{4}} ((1 + x^{\frac{1}{4}}) (1 - 1 x^{\frac{1}{4}} + {(x^{\frac{1}{4}})}^{2}))$

Step 5.1.2

Rewrite $- 1 x^{\frac{1}{4}}$ as $- x^{\frac{1}{4}}$ .

$x^{\frac{7}{4}} ((1 + x^{\frac{1}{4}}) (1 - x^{\frac{1}{4}} + {(x^{\frac{1}{4}})}^{2}))$

Step 5.1.3

Multiply the exponents in ${(x^{\frac{1}{4}})}^{2}$ .

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Step 5.1.3.1

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

$x^{\frac{7}{4}} ((1 + x^{\frac{1}{4}}) (1 - x^{\frac{1}{4}} + x^{\frac{1}{4} \cdot 2}))$

Step 5.1.3.2

Cancel the common factor of $2$ .

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Step 5.1.3.2.1

Factor $2$ out of $4$ .

$x^{\frac{7}{4}} ((1 + x^{\frac{1}{4}}) (1 - x^{\frac{1}{4}} + x^{\frac{1}{2 (2)} \cdot 2}))$

Step 5.1.3.2.2

Cancel the common factor.

$x^{\frac{7}{4}} ((1 + x^{\frac{1}{4}}) (1 - x^{\frac{1}{4}} + x^{\frac{1}{2 \cdot 2} \cdot 2}))$

Step 5.1.3.2.3

Rewrite the expression.

$x^{\frac{7}{4}} ((1 + x^{\frac{1}{4}}) (1 - x^{\frac{1}{4}} + x^{\frac{1}{2}}))$

Step 5.2

Remove unnecessary parentheses.

$x^{\frac{7}{4}} (1 + x^{\frac{1}{4}}) (1 - x^{\frac{1}{4}} + x^{\frac{1}{2}})$