Algebra Examples

$\frac{3}{\sqrt{x}} + \frac{x}{\sqrt[3]{y}}$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{3}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{3}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}} + \frac{x}{\sqrt[3]{y}}$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{3}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{3 \sqrt{x}}{\sqrt{x} \sqrt{x}} + \frac{x}{\sqrt[3]{y}}$

Step 1.2.2

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

$\frac{3 \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}} + \frac{x}{\sqrt[3]{y}}$

Step 1.2.3

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

$\frac{3 \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}} + \frac{x}{\sqrt[3]{y}}$

Step 1.2.4

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

$\frac{3 \sqrt{x}}{{\sqrt{x}}^{1 + 1}} + \frac{x}{\sqrt[3]{y}}$

Step 1.2.5

Add $1$ and $1$ .

$\frac{3 \sqrt{x}}{{\sqrt{x}}^{2}} + \frac{x}{\sqrt[3]{y}}$

Step 1.2.6

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

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Step 1.2.6.1

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

$\frac{3 \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}} + \frac{x}{\sqrt[3]{y}}$

Step 1.2.6.2

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

$\frac{3 \sqrt{x}}{x^{\frac{1}{2} \cdot 2}} + \frac{x}{\sqrt[3]{y}}$

Step 1.2.6.3

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

$\frac{3 \sqrt{x}}{x^{\frac{2}{2}}} + \frac{x}{\sqrt[3]{y}}$

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

$\frac{3 \sqrt{x}}{x^{\frac{2}{2}}} + \frac{x}{\sqrt[3]{y}}$

Step 1.2.6.4.2

Rewrite the expression.

$\frac{3 \sqrt{x}}{x^{1}} + \frac{x}{\sqrt[3]{y}}$

Step 1.2.6.5

Simplify.

$\frac{3 \sqrt{x}}{x} + \frac{x}{\sqrt[3]{y}}$

Step 1.3

Multiply $\frac{x}{\sqrt[3]{y}}$ by $\frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}}$ .

$\frac{3 \sqrt{x}}{x} + \frac{x}{\sqrt[3]{y}} \cdot \frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}}$

Step 1.4

Combine and simplify the denominator.

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Step 1.4.1

Multiply $\frac{x}{\sqrt[3]{y}}$ by $\frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}}$ .

$\frac{3 \sqrt{x}}{x} + \frac{x {\sqrt[3]{y}}^{2}}{\sqrt[3]{y} {\sqrt[3]{y}}^{2}}$

Step 1.4.2

Raise $\sqrt[3]{y}$ to the power of $1$ .

$\frac{3 \sqrt{x}}{x} + \frac{x {\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{1} {\sqrt[3]{y}}^{2}}$

Step 1.4.3

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

$\frac{3 \sqrt{x}}{x} + \frac{x {\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{1 + 2}}$

Step 1.4.4

Add $1$ and $2$ .

$\frac{3 \sqrt{x}}{x} + \frac{x {\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{3}}$

Step 1.4.5

Rewrite ${\sqrt[3]{y}}^{3}$ as $y$ .

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Step 1.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y}$ as $y^{\frac{1}{3}}$ .

$\frac{3 \sqrt{x}}{x} + \frac{x {\sqrt[3]{y}}^{2}}{{(y^{\frac{1}{3}})}^{3}}$

Step 1.4.5.2

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

$\frac{3 \sqrt{x}}{x} + \frac{x {\sqrt[3]{y}}^{2}}{y^{\frac{1}{3} \cdot 3}}$

Step 1.4.5.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{3 \sqrt{x}}{x} + \frac{x {\sqrt[3]{y}}^{2}}{y^{\frac{3}{3}}}$

Step 1.4.5.4

Cancel the common factor of $3$ .

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Step 1.4.5.4.1

Cancel the common factor.

$\frac{3 \sqrt{x}}{x} + \frac{x {\sqrt[3]{y}}^{2}}{y^{\frac{3}{3}}}$

Step 1.4.5.4.2

Rewrite the expression.

$\frac{3 \sqrt{x}}{x} + \frac{x {\sqrt[3]{y}}^{2}}{y^{1}}$

Step 1.4.5.5

Simplify.

$\frac{3 \sqrt{x}}{x} + \frac{x {\sqrt[3]{y}}^{2}}{y}$

Step 1.5

Rewrite ${\sqrt[3]{y}}^{2}$ as $\sqrt[3]{y^{2}}$ .

$\frac{3 \sqrt{x}}{x} + \frac{x \sqrt[3]{y^{2}}}{y}$

Step 2

To write $\frac{3 \sqrt{x}}{x}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{3 \sqrt{x}}{x} \cdot \frac{y}{y} + \frac{x \sqrt[3]{y^{2}}}{y}$

Step 3

To write $\frac{x \sqrt[3]{y^{2}}}{y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{3 \sqrt{x}}{x} \cdot \frac{y}{y} + \frac{x \sqrt[3]{y^{2}}}{y} \cdot \frac{x}{x}$

Step 4

Write each expression with a common denominator of $x y$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{3 \sqrt{x}}{x}$ by $\frac{y}{y}$ .

$\frac{3 \sqrt{x} y}{x y} + \frac{x \sqrt[3]{y^{2}}}{y} \cdot \frac{x}{x}$

Step 4.2

Multiply $\frac{x \sqrt[3]{y^{2}}}{y}$ by $\frac{x}{x}$ .

$\frac{3 \sqrt{x} y}{x y} + \frac{x \sqrt[3]{y^{2}} x}{y x}$

Step 4.3

Reorder the factors of $y x$ .

$\frac{3 \sqrt{x} y}{x y} + \frac{x \sqrt[3]{y^{2}} x}{x y}$

Step 5

Combine the numerators over the common denominator.

$\frac{3 \sqrt{x} y + x \sqrt[3]{y^{2}} x}{x y}$

Step 6

Simplify the numerator.

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Step 6.1

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

$\frac{3 x^{\frac{1}{2}} y + x \sqrt[3]{y^{2}} x}{x y}$

Step 6.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y^{2}}$ as $y^{\frac{2}{3}}$ .

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

Step 6.3

Factor $x^{\frac{1}{2}} y^{\frac{2}{3}}$ out of $3 x^{\frac{1}{2}} y + x y^{\frac{2}{3}} x$ .

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Step 6.3.1

Move $y^{\frac{2}{3}}$ .

$\frac{3 x^{\frac{1}{2}} y + x \cdot x y^{\frac{2}{3}}}{x y}$

Step 6.3.2

Factor $x^{\frac{1}{2}} y^{\frac{2}{3}}$ out of $3 x^{\frac{1}{2}} y$ .

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

Step 6.3.3

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

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

Step 6.3.4

Factor $x^{\frac{1}{2}} y^{\frac{2}{3}}$ out of $x^{\frac{1}{2}} y^{\frac{2}{3}} (3 y^{\frac{1}{3}}) + x^{\frac{1}{2}} y^{\frac{2}{3}} (x^{\frac{1}{2}} x)$ .

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

Step 6.4

Multiply $x^{\frac{1}{2}}$ by $x$ by adding the exponents.

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Step 6.4.1

Multiply $x^{\frac{1}{2}}$ by $x$ .

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Step 6.4.1.1

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

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

Step 6.4.1.2

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

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

Step 6.4.2

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

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

Step 6.4.3

Combine the numerators over the common denominator.

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

Step 6.4.4

Add $1$ and $2$ .

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

Step 7

Simplify.

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Step 7.1

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 7.2

Move $y^{\frac{2}{3}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 8

Simplify the denominator.

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Step 8.1

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 8.1.1

Move $x^{- \frac{1}{2}}$ .

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

Step 8.1.2

Multiply $x^{- \frac{1}{2}}$ by $x$ .

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Step 8.1.2.1

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

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

Step 8.1.2.2

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

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

Step 8.1.3

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

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

Step 8.1.4

Combine the numerators over the common denominator.

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

Step 8.1.5

Add $- 1$ and $2$ .

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

Step 8.2

Multiply $y$ by $y^{- \frac{2}{3}}$ by adding the exponents.

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Step 8.2.1

Move $y^{- \frac{2}{3}}$ .

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

Step 8.2.2

Multiply $y^{- \frac{2}{3}}$ by $y$ .

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Step 8.2.2.1

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

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

Step 8.2.2.2

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

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

Step 8.2.3

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

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

Step 8.2.4

Combine the numerators over the common denominator.

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

Step 8.2.5

Add $- 2$ and $3$ .

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