Basic Math Examples

$\frac{3 a^{2} \sqrt{98 a^{5} y^{3}}}{\sqrt{49 a^{4} y}} + \frac{4 a y \sqrt{50 a^{3 y}}}{5 \sqrt{y}}$

Step 1

Simplify each term.

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Step 1.1

Combine $\sqrt{98 a^{5} y^{3}}$ and $\sqrt{49 a^{4} y}$ into a single radical.

$3 a^{2} \sqrt{\frac{98 a^{5} y^{3}}{49 a^{4} y}} + \frac{4 a y \sqrt{50 a^{3 y}}}{5 \sqrt{y}}$

Step 1.2

Reduce the expression $\frac{98 a^{5} y^{3}}{49 a^{4} y}$ by cancelling the common factors.

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Step 1.2.1

Factor $49$ out of $98 a^{5} y^{3}$ .

$3 a^{2} \sqrt{\frac{49 (2 a^{5} y^{3})}{49 a^{4} y}} + \frac{4 a y \sqrt{50 a^{3 y}}}{5 \sqrt{y}}$

Step 1.2.2

Factor $49$ out of $49 a^{4} y$ .

$3 a^{2} \sqrt{\frac{49 (2 a^{5} y^{3})}{49 (a^{4} y)}} + \frac{4 a y \sqrt{50 a^{3 y}}}{5 \sqrt{y}}$

Step 1.2.3

Cancel the common factor.

$3 a^{2} \sqrt{\frac{49 (2 a^{5} y^{3})}{49 (a^{4} y)}} + \frac{4 a y \sqrt{50 a^{3 y}}}{5 \sqrt{y}}$

Step 1.2.4

Rewrite the expression.

$3 a^{2} \sqrt{\frac{2 a^{5} y^{3}}{a^{4} y}} + \frac{4 a y \sqrt{50 a^{3 y}}}{5 \sqrt{y}}$

Step 1.3

Cancel the common factor of $a^{5}$ and $a^{4}$ .

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Step 1.3.1

Factor $a^{4}$ out of $2 a^{5} y^{3}$ .

$3 a^{2} \sqrt{\frac{a^{4} (2 a y^{3})}{a^{4} y}} + \frac{4 a y \sqrt{50 a^{3 y}}}{5 \sqrt{y}}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $a^{4}$ out of $a^{4} y$ .

$3 a^{2} \sqrt{\frac{a^{4} (2 a y^{3})}{a^{4} (y)}} + \frac{4 a y \sqrt{50 a^{3 y}}}{5 \sqrt{y}}$

Step 1.3.2.2

Cancel the common factor.

$3 a^{2} \sqrt{\frac{a^{4} (2 a y^{3})}{a^{4} y}} + \frac{4 a y \sqrt{50 a^{3 y}}}{5 \sqrt{y}}$

Step 1.3.2.3

Rewrite the expression.

$3 a^{2} \sqrt{\frac{2 a y^{3}}{y}} + \frac{4 a y \sqrt{50 a^{3 y}}}{5 \sqrt{y}}$

Step 1.4

Cancel the common factor of $y^{3}$ and $y$ .

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Step 1.4.1

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

$3 a^{2} \sqrt{\frac{y (2 a y^{2})}{y}} + \frac{4 a y \sqrt{50 a^{3 y}}}{5 \sqrt{y}}$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

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

$3 a^{2} \sqrt{\frac{y (2 a y^{2})}{y^{1}}} + \frac{4 a y \sqrt{50 a^{3 y}}}{5 \sqrt{y}}$

Step 1.4.2.2

Factor $y$ out of $y^{1}$ .

$3 a^{2} \sqrt{\frac{y (2 a y^{2})}{y \cdot 1}} + \frac{4 a y \sqrt{50 a^{3 y}}}{5 \sqrt{y}}$

Step 1.4.2.3

Cancel the common factor.

$3 a^{2} \sqrt{\frac{y (2 a y^{2})}{y \cdot 1}} + \frac{4 a y \sqrt{50 a^{3 y}}}{5 \sqrt{y}}$

Step 1.4.2.4

Rewrite the expression.

$3 a^{2} \sqrt{\frac{2 a y^{2}}{1}} + \frac{4 a y \sqrt{50 a^{3 y}}}{5 \sqrt{y}}$

Step 1.4.2.5

Divide $2 a y^{2}$ by $1$ .

$3 a^{2} \sqrt{2 a y^{2}} + \frac{4 a y \sqrt{50 a^{3 y}}}{5 \sqrt{y}}$

Step 1.5

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

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Step 1.5.1

Move $a$ .

$3 a^{2} \sqrt{2 y^{2} a} + \frac{4 a y \sqrt{50 a^{3 y}}}{5 \sqrt{y}}$

Step 1.5.2

Reorder $2$ and $y^{2}$ .

$3 a^{2} \sqrt{y^{2} \cdot 2 a} + \frac{4 a y \sqrt{50 a^{3 y}}}{5 \sqrt{y}}$

Step 1.5.3

Add parentheses.

$3 a^{2} \sqrt{y^{2} \cdot (2 a)} + \frac{4 a y \sqrt{50 a^{3 y}}}{5 \sqrt{y}}$

Step 1.6

Pull terms out from under the radical.

$3 a^{2} (y \sqrt{2 a}) + \frac{4 a y \sqrt{50 a^{3 y}}}{5 \sqrt{y}}$

Step 1.7

Simplify the numerator.

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Step 1.7.1

Rewrite $50 a^{3 y}$ as $5^{2} \cdot (2 a^{3 y})$ .

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Step 1.7.1.1

Factor $25$ out of $50$ .

$3 a^{2} (y \sqrt{2 a}) + \frac{4 a y \sqrt{25 (2) a^{3 y}}}{5 \sqrt{y}}$

Step 1.7.1.2

Rewrite $25$ as $5^{2}$ .

$3 a^{2} (y \sqrt{2 a}) + \frac{4 a y \sqrt{5^{2} \cdot 2 a^{3 y}}}{5 \sqrt{y}}$

Step 1.7.1.3

Add parentheses.

$3 a^{2} (y \sqrt{2 a}) + \frac{4 a y \sqrt{5^{2} \cdot (2 a^{3 y})}}{5 \sqrt{y}}$

Step 1.7.2

Pull terms out from under the radical.

$3 a^{2} (y \sqrt{2 a}) + \frac{4 a y \cdot 5 \sqrt{2 a^{3 y}}}{5 \sqrt{y}}$

Step 1.7.3

Multiply $5$ by $4$ .

$3 a^{2} (y \sqrt{2 a}) + \frac{20 a y \sqrt{2 a^{3 y}}}{5 \sqrt{y}}$

Step 1.8

Cancel the common factor of $20$ and $5$ .

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Step 1.8.1

Factor $5$ out of $20 a y \sqrt{2 a^{3 y}}$ .

$3 a^{2} (y \sqrt{2 a}) + \frac{5 (4 a y \sqrt{2 a^{3 y}})}{5 \sqrt{y}}$

Step 1.8.2

Cancel the common factors.

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Step 1.8.2.1

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

$3 a^{2} (y \sqrt{2 a}) + \frac{5 (4 a y \sqrt{2 a^{3 y}})}{5 (\sqrt{y})}$

Step 1.8.2.2

Cancel the common factor.

$3 a^{2} (y \sqrt{2 a}) + \frac{5 (4 a y \sqrt{2 a^{3 y}})}{5 \sqrt{y}}$

Step 1.8.2.3

Rewrite the expression.

$3 a^{2} (y \sqrt{2 a}) + \frac{4 a y \sqrt{2 a^{3 y}}}{\sqrt{y}}$

Step 1.9

Multiply $\frac{4 a y \sqrt{2 a^{3 y}}}{\sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

$3 a^{2} (y \sqrt{2 a}) + \frac{4 a y \sqrt{2 a^{3 y}}}{\sqrt{y}} \cdot \frac{\sqrt{y}}{\sqrt{y}}$

Step 1.10

Combine and simplify the denominator.

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Step 1.10.1

Multiply $\frac{4 a y \sqrt{2 a^{3 y}}}{\sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

$3 a^{2} (y \sqrt{2 a}) + \frac{4 a y \sqrt{2 a^{3 y}} \sqrt{y}}{\sqrt{y} \sqrt{y}}$

Step 1.10.2

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

$3 a^{2} (y \sqrt{2 a}) + \frac{4 a y \sqrt{2 a^{3 y}} \sqrt{y}}{{\sqrt{y}}^{1} \sqrt{y}}$

Step 1.10.3

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

$3 a^{2} (y \sqrt{2 a}) + \frac{4 a y \sqrt{2 a^{3 y}} \sqrt{y}}{{\sqrt{y}}^{1} {\sqrt{y}}^{1}}$

Step 1.10.4

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

$3 a^{2} (y \sqrt{2 a}) + \frac{4 a y \sqrt{2 a^{3 y}} \sqrt{y}}{{\sqrt{y}}^{1 + 1}}$

Step 1.10.5

Add $1$ and $1$ .

$3 a^{2} (y \sqrt{2 a}) + \frac{4 a y \sqrt{2 a^{3 y}} \sqrt{y}}{{\sqrt{y}}^{2}}$

Step 1.10.6

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

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Step 1.10.6.1

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

$3 a^{2} (y \sqrt{2 a}) + \frac{4 a y \sqrt{2 a^{3 y}} \sqrt{y}}{{(y^{\frac{1}{2}})}^{2}}$

Step 1.10.6.2

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

$3 a^{2} (y \sqrt{2 a}) + \frac{4 a y \sqrt{2 a^{3 y}} \sqrt{y}}{y^{\frac{1}{2} \cdot 2}}$

Step 1.10.6.3

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

$3 a^{2} (y \sqrt{2 a}) + \frac{4 a y \sqrt{2 a^{3 y}} \sqrt{y}}{y^{\frac{2}{2}}}$

Step 1.10.6.4

Cancel the common factor of $2$ .

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Step 1.10.6.4.1

Cancel the common factor.

$3 a^{2} (y \sqrt{2 a}) + \frac{4 a y \sqrt{2 a^{3 y}} \sqrt{y}}{y^{\frac{2}{2}}}$

Step 1.10.6.4.2

Rewrite the expression.

$3 a^{2} (y \sqrt{2 a}) + \frac{4 a y \sqrt{2 a^{3 y}} \sqrt{y}}{y^{1}}$

Step 1.10.6.5

Simplify.

$3 a^{2} (y \sqrt{2 a}) + \frac{4 a y \sqrt{2 a^{3 y}} \sqrt{y}}{y}$

Step 1.11

Cancel the common factor of $y$ .

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Step 1.11.1

Cancel the common factor.

$3 a^{2} (y \sqrt{2 a}) + \frac{4 a y \sqrt{2 a^{3 y}} \sqrt{y}}{y}$

Step 1.11.2

Divide $4 a \sqrt{2 a^{3 y}} \sqrt{y}$ by $1$ .

$3 a^{2} (y \sqrt{2 a}) + 4 a \sqrt{2 a^{3 y}} \sqrt{y}$

Step 1.12

Combine using the product rule for radicals.

$3 a^{2} (y \sqrt{2 a}) + 4 a \sqrt{y \cdot 2 a^{3 y}}$

Step 2

Reorder factors in $3 a^{2} (y \sqrt{2 a}) + 4 a \sqrt{y \cdot 2 a^{3 y}}$ .

$3 a^{2} y \sqrt{2 a} + 4 a \sqrt{2 y a^{3 y}}$