Algebra Examples

$x^{\frac{3}{2}} + 125 y^{\frac{9}{2}}$

Step 1

Rewrite $x^{\frac{3}{2}}$ as ${(x^{\frac{1}{2}})}^{3}$ .

${(x^{\frac{1}{2}})}^{3} + 125 y^{\frac{9}{2}}$

Step 2

Rewrite $125 y^{\frac{9}{2}}$ as ${(5 y^{\frac{3}{2}})}^{3}$ .

${(x^{\frac{1}{2}})}^{3} + {(5 y^{\frac{3}{2}})}^{3}$

Step 3

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 $a = x^{\frac{1}{2}}$ and $b = 5 y^{\frac{3}{2}}$ .

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

Step 4

Simplify.

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Step 4.1

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

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Step 4.1.1

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

$(x^{\frac{1}{2}} + 5 y^{\frac{3}{2}}) (x^{\frac{1}{2} \cdot 2} - x^{\frac{1}{2}} (5 y^{\frac{3}{2}}) + {(5 y^{\frac{3}{2}})}^{2})$

Step 4.1.2

Cancel the common factor of $2$ .

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Step 4.1.2.1

Cancel the common factor.

$(x^{\frac{1}{2}} + 5 y^{\frac{3}{2}}) (x^{\frac{1}{2} \cdot 2} - x^{\frac{1}{2}} (5 y^{\frac{3}{2}}) + {(5 y^{\frac{3}{2}})}^{2})$

Step 4.1.2.2

Rewrite the expression.

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

Step 4.2

Simplify.

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

Step 4.3

Rewrite using the commutative property of multiplication.

$(x^{\frac{1}{2}} + 5 y^{\frac{3}{2}}) (x - 1 \cdot 5 x^{\frac{1}{2}} y^{\frac{3}{2}} + {(5 y^{\frac{3}{2}})}^{2})$

Step 4.4

Multiply $- 1$ by $5$ .

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

Step 4.5

Apply the product rule to $5 y^{\frac{3}{2}}$ .

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

Step 4.6

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

$(x^{\frac{1}{2}} + 5 y^{\frac{3}{2}}) (x - 5 x^{\frac{1}{2}} y^{\frac{3}{2}} + 25 {(y^{\frac{3}{2}})}^{2})$

Step 4.7

Multiply the exponents in ${(y^{\frac{3}{2}})}^{2}$ .

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Step 4.7.1

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

$(x^{\frac{1}{2}} + 5 y^{\frac{3}{2}}) (x - 5 x^{\frac{1}{2}} y^{\frac{3}{2}} + 25 y^{\frac{3}{2} \cdot 2})$

Step 4.7.2

Cancel the common factor of $2$ .

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Step 4.7.2.1

Cancel the common factor.

$(x^{\frac{1}{2}} + 5 y^{\frac{3}{2}}) (x - 5 x^{\frac{1}{2}} y^{\frac{3}{2}} + 25 y^{\frac{3}{2} \cdot 2})$

Step 4.7.2.2

Rewrite the expression.

$(x^{\frac{1}{2}} + 5 y^{\frac{3}{2}}) (x - 5 x^{\frac{1}{2}} y^{\frac{3}{2}} + 25 y^{3})$