Algebra Examples

$\frac{4 \sqrt{15 y^{6}} - 18 \sqrt{45 y^{5}}}{2 \sqrt{5 y^{4}}} y^{3}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $2$ out of $4 \sqrt{15 y^{6}} - 18 \sqrt{45 y^{5}}$ .

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Step 1.1.1

Factor $2$ out of $4 \sqrt{15 y^{6}}$ .

$\frac{2 (2 \sqrt{15 y^{6}}) - 18 \sqrt{45 y^{5}}}{2 \sqrt{5 y^{4}}} \cdot y^{3}$

Step 1.1.2

Factor $2$ out of $- 18 \sqrt{45 y^{5}}$ .

$\frac{2 (2 \sqrt{15 y^{6}}) + 2 (- 9 \sqrt{45 y^{5}})}{2 \sqrt{5 y^{4}}} \cdot y^{3}$

Step 1.1.3

Factor $2$ out of $2 (2 \sqrt{15 y^{6}}) + 2 (- 9 \sqrt{45 y^{5}})$ .

$\frac{2 (2 \sqrt{15 y^{6}} - 9 \sqrt{45 y^{5}})}{2 \sqrt{5 y^{4}}} \cdot y^{3}$

Step 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15 y^{6}}$ as ${(15 y^{6})}^{\frac{1}{2}}$ .

$\frac{2 (2 {(15 y^{6})}^{\frac{1}{2}} - 9 \sqrt{45 y^{5}})}{2 \sqrt{5 y^{4}}} \cdot y^{3}$

Step 1.3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{45 y^{5}}$ as ${(45 y^{5})}^{\frac{1}{2}}$ .

$\frac{2 (2 {(15 y^{6})}^{\frac{1}{2}} - 9 {(45 y^{5})}^{\frac{1}{2}})}{2 \sqrt{5 y^{4}}} \cdot y^{3}$

Step 1.4

Apply the product rule to $15 y^{6}$ .

$\frac{2 (2 (15^{\frac{1}{2}} {(y^{6})}^{\frac{1}{2}}) - 9 {(45 y^{5})}^{\frac{1}{2}})}{2 \sqrt{5 y^{4}}} \cdot y^{3}$

Step 1.5

Multiply the exponents in ${(y^{6})}^{\frac{1}{2}}$ .

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Step 1.5.1

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

$\frac{2 (2 (15^{\frac{1}{2}} y^{6 (\frac{1}{2})}) - 9 {(45 y^{5})}^{\frac{1}{2}})}{2 \sqrt{5 y^{4}}} \cdot y^{3}$

Step 1.5.2

Cancel the common factor of $2$ .

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Step 1.5.2.1

Factor $2$ out of $6$ .

$\frac{2 (2 (15^{\frac{1}{2}} y^{2 (3) \frac{1}{2}}) - 9 {(45 y^{5})}^{\frac{1}{2}})}{2 \sqrt{5 y^{4}}} \cdot y^{3}$

Step 1.5.2.2

Cancel the common factor.

$\frac{2 (2 (15^{\frac{1}{2}} y^{2 \cdot 3 \frac{1}{2}}) - 9 {(45 y^{5})}^{\frac{1}{2}})}{2 \sqrt{5 y^{4}}} \cdot y^{3}$

Step 1.5.2.3

Rewrite the expression.

$\frac{2 (2 (15^{\frac{1}{2}} y^{3}) - 9 {(45 y^{5})}^{\frac{1}{2}})}{2 \sqrt{5 y^{4}}} \cdot y^{3}$

Step 1.6

Apply the product rule to $45 y^{5}$ .

$\frac{2 (2 \cdot 15^{\frac{1}{2}} y^{3} - 9 (45^{\frac{1}{2}} {(y^{5})}^{\frac{1}{2}}))}{2 \sqrt{5 y^{4}}} \cdot y^{3}$

Step 1.7

Multiply the exponents in ${(y^{5})}^{\frac{1}{2}}$ .

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Step 1.7.1

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

$\frac{2 (2 \cdot 15^{\frac{1}{2}} y^{3} - 9 (45^{\frac{1}{2}} y^{5 (\frac{1}{2})}))}{2 \sqrt{5 y^{4}}} \cdot y^{3}$

Step 1.7.2

Combine $5$ and $\frac{1}{2}$ .

$\frac{2 (2 \cdot 15^{\frac{1}{2}} y^{3} - 9 (45^{\frac{1}{2}} y^{\frac{5}{2}}))}{2 \sqrt{5 y^{4}}} \cdot y^{3}$

Step 1.8

Factor $y^{\frac{5}{2}}$ out of $2 \cdot 15^{\frac{1}{2}} y^{3} - 9 \cdot 45^{\frac{1}{2}} y^{\frac{5}{2}}$ .

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Step 1.8.1

Factor $y^{\frac{5}{2}}$ out of $2 \cdot 15^{\frac{1}{2}} y^{3}$ .

$\frac{2 (y^{\frac{5}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}}) - 9 \cdot 45^{\frac{1}{2}} y^{\frac{5}{2}})}{2 \sqrt{5 y^{4}}} \cdot y^{3}$

Step 1.8.2

Factor $y^{\frac{5}{2}}$ out of $- 9 \cdot 45^{\frac{1}{2}} y^{\frac{5}{2}}$ .

$\frac{2 (y^{\frac{5}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}}) + y^{\frac{5}{2}} (- 9 \cdot 45^{\frac{1}{2}}))}{2 \sqrt{5 y^{4}}} \cdot y^{3}$

Step 1.8.3

Factor $y^{\frac{5}{2}}$ out of $y^{\frac{5}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}}) + y^{\frac{5}{2}} (- 9 \cdot 45^{\frac{1}{2}})$ .

$\frac{2 (y^{\frac{5}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}}))}{2 \sqrt{5 y^{4}}} \cdot y^{3}$

$\frac{2 y^{\frac{5}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}})}{2 \sqrt{5 y^{4}}} \cdot y^{3}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $5 y^{4}$ as ${(y^{2})}^{2} \cdot 5$ .

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Step 2.1.1

Rewrite $y^{4}$ as ${(y^{2})}^{2}$ .

$\frac{2 y^{\frac{5}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}})}{2 \sqrt{5 {(y^{2})}^{2}}} \cdot y^{3}$

Step 2.1.2

Reorder $5$ and ${(y^{2})}^{2}$ .

$\frac{2 y^{\frac{5}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}})}{2 \sqrt{{(y^{2})}^{2} \cdot 5}} \cdot y^{3}$

Step 2.2

Pull terms out from under the radical.

$\frac{2 y^{\frac{5}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}})}{2 y^{2} \sqrt{5}} \cdot y^{3}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $y^{2}$ .

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Step 3.1.1

Factor $y^{2}$ out of $2 y^{2} \sqrt{5}$ .

$\frac{2 y^{\frac{5}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}})}{y^{2} (2 \sqrt{5})} \cdot y^{3}$

Step 3.1.2

Factor $y^{2}$ out of $y^{3}$ .

$\frac{2 y^{\frac{5}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}})}{y^{2} (2 \sqrt{5})} \cdot (y^{2} y)$

Step 3.1.3

Cancel the common factor.

$\frac{2 y^{\frac{5}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}})}{y^{2} (2 \sqrt{5})} \cdot (y^{2} y)$

Step 3.1.4

Rewrite the expression.

$\frac{2 y^{\frac{5}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}})}{2 \sqrt{5}} \cdot y$

Step 3.2

Combine $\frac{2 y^{\frac{5}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}})}{2 \sqrt{5}}$ and $y$ .

$\frac{2 y^{\frac{5}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}}) y}{2 \sqrt{5}}$

Step 4

Raise $y$ to the power of $1$ .

$\frac{2 (y^{1} y^{\frac{5}{2}}) (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}})}{2 \sqrt{5}}$

Step 5

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

$\frac{2 y^{1 + \frac{5}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}})}{2 \sqrt{5}}$

Step 6

Simplify the expression.

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Step 6.1

Write $1$ as a fraction with a common denominator.

$\frac{2 y^{\frac{2}{2} + \frac{5}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}})}{2 \sqrt{5}}$

Step 6.2

Combine the numerators over the common denominator.

$\frac{2 y^{\frac{2 + 5}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}})}{2 \sqrt{5}}$

Step 6.3

Add $2$ and $5$ .

$\frac{2 y^{\frac{7}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}})}{2 \sqrt{5}}$

Step 7

Cancel the common factor.

$\frac{2 y^{\frac{7}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}})}{2 \sqrt{5}}$

Step 8

Rewrite the expression.

$\frac{y^{\frac{7}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}})}{\sqrt{5}}$

Step 9

Multiply $\frac{y^{\frac{7}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}})}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{y^{\frac{7}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}})}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{y^{\frac{7}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}})}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{y^{\frac{7}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}}) \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 10.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{y^{\frac{7}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}}) \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 10.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{y^{\frac{7}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}}) \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 10.4

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

$\frac{y^{\frac{7}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}}) \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 10.5

Add $1$ and $1$ .

$\frac{y^{\frac{7}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}}) \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 10.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 10.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{y^{\frac{7}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}}) \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 10.6.2

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

$\frac{y^{\frac{7}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}}) \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 10.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{y^{\frac{7}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}}) \sqrt{5}}{5^{\frac{2}{2}}}$

Step 10.6.4

Cancel the common factor of $2$ .

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Step 10.6.4.1

Cancel the common factor.

$\frac{y^{\frac{7}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}}) \sqrt{5}}{5^{\frac{2}{2}}}$

Step 10.6.4.2

Rewrite the expression.

$\frac{y^{\frac{7}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}}) \sqrt{5}}{5^{1}}$

Step 10.6.5

Evaluate the exponent.

$\frac{y^{\frac{7}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}}) \sqrt{5}}{5}$

Step 11

Reorder factors in $\frac{y^{\frac{7}{2}} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}}) \sqrt{5}}{5}$ .

$\frac{y^{\frac{7}{2}} \sqrt{5} (2 \cdot 15^{\frac{1}{2}} y^{\frac{1}{2}} - 9 \cdot 45^{\frac{1}{2}})}{5}$