Precalculus Examples

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

Step 1

Expand $(x^{\frac{7}{2}} + y^{\frac{7}{2}}) (x^{\frac{7}{2}} - y^{\frac{7}{2}})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $x^{\frac{7}{2}} x^{\frac{7}{2}} + x^{\frac{7}{2}} (- y^{\frac{7}{2}}) + y^{\frac{7}{2}} x^{\frac{7}{2}} + y^{\frac{7}{2}} (- y^{\frac{7}{2}})$ .

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Step 2.1.1

Reorder the factors in the terms $x^{\frac{7}{2}} (- y^{\frac{7}{2}})$ and $y^{\frac{7}{2}} x^{\frac{7}{2}}$ .

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

Step 2.1.2

Add $- x^{\frac{7}{2}} y^{\frac{7}{2}}$ and $x^{\frac{7}{2}} y^{\frac{7}{2}}$ .

$x^{\frac{7}{2}} x^{\frac{7}{2}} + 0 + y^{\frac{7}{2}} (- y^{\frac{7}{2}})$

Step 2.1.3

Add $x^{\frac{7}{2}} x^{\frac{7}{2}}$ and $0$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $x^{\frac{7}{2}}$ by $x^{\frac{7}{2}}$ by adding the exponents.

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Step 2.2.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.2.1.2

Combine the numerators over the common denominator.

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

Step 2.2.1.3

Add $7$ and $7$ .

$x^{\frac{14}{2}} + y^{\frac{7}{2}} (- y^{\frac{7}{2}})$

Step 2.2.1.4

Divide $14$ by $2$ .

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

Step 2.2.2

Rewrite using the commutative property of multiplication.

$x^{7} - y^{\frac{7}{2}} y^{\frac{7}{2}}$

Step 2.2.3

Multiply $y^{\frac{7}{2}}$ by $y^{\frac{7}{2}}$ by adding the exponents.

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Step 2.2.3.1

Move $y^{\frac{7}{2}}$ .

$x^{7} - (y^{\frac{7}{2}} y^{\frac{7}{2}})$

Step 2.2.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{7} - y^{\frac{7}{2} + \frac{7}{2}}$

Step 2.2.3.3

Combine the numerators over the common denominator.

$x^{7} - y^{\frac{7 + 7}{2}}$

Step 2.2.3.4

Add $7$ and $7$ .

$x^{7} - y^{\frac{14}{2}}$

Step 2.2.3.5

Divide $14$ by $2$ .

$x^{7} - y^{7}$