Finite Math Examples

$\sqrt{4 x^{2} y^{5}} + \sqrt{x^{3} y^{5}} - 5 x y \sqrt{x^{5} y^{3}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x^{2} y^{5}}$ as ${(4 x^{2} y^{5})}^{\frac{1}{2}}$ .

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3} y^{5}}$ as ${(x^{3} y^{5})}^{\frac{1}{2}}$ .

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

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{5} y^{3}}$ as ${(x^{5} y^{3})}^{\frac{1}{2}}$ .

${(4 x^{2} y^{5})}^{\frac{1}{2}} + {(x^{3} y^{5})}^{\frac{1}{2}} - 5 x y {(x^{5} y^{3})}^{\frac{1}{2}}$

Step 4

Use the power rule ${(a i)}^{n} = a^{n} i^{n}$ to distribute the exponent.

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Step 4.1

Apply the product rule to $4 x^{2} y^{5}$ .

${(4 x^{2})}^{\frac{1}{2}} {(y^{5})}^{\frac{1}{2}} + {(x^{3} y^{5})}^{\frac{1}{2}} - 5 x y {(x^{5} y^{3})}^{\frac{1}{2}}$

Step 4.2

Apply the product rule to $4 x^{2}$ .

$4^{\frac{1}{2}} {(x^{2})}^{\frac{1}{2}} {(y^{5})}^{\frac{1}{2}} + {(x^{3} y^{5})}^{\frac{1}{2}} - 5 x y {(x^{5} y^{3})}^{\frac{1}{2}}$

Step 5

Rewrite $4$ as $2^{2}$ .

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

Step 6

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

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

Step 7

Cancel the common factor of $2$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

Evaluate the exponent.

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

Step 9

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

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Step 9.1

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

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

Step 9.2

Cancel the common factor of $2$ .

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Step 9.2.1

Cancel the common factor.

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

Step 9.2.2

Rewrite the expression.

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

Step 10

Simplify.

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

Step 11

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

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Step 11.1

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

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

Step 11.2

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

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

Step 12

Apply the product rule to $x^{3} y^{5}$ .

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

Step 13

Multiply the exponents in ${(x^{3})}^{\frac{1}{2}}$ .

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Step 13.1

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

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

Step 13.2

Combine $3$ and $\frac{1}{2}$ .

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

Step 14

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

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Step 14.1

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

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

Step 14.2

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

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

Step 15

Apply the product rule to $x^{5} y^{3}$ .

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

Step 16

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

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Step 16.1

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

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

Step 16.2

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

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

Step 17

Multiply $x$ by $x^{\frac{5}{2}}$ by adding the exponents.

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Step 17.1

Move $x^{\frac{5}{2}}$ .

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

Step 17.2

Multiply $x^{\frac{5}{2}}$ by $x$ .

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Step 17.2.1

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

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

Step 17.2.2

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

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

Step 17.3

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

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

Step 17.4

Combine the numerators over the common denominator.

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

Step 17.5

Add $5$ and $2$ .

$2 x y^{\frac{5}{2}} + x^{\frac{3}{2}} y^{\frac{5}{2}} - 5 x^{\frac{7}{2}} y {(y^{3})}^{\frac{1}{2}}$

Step 18

Multiply the exponents in ${(y^{3})}^{\frac{1}{2}}$ .

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Step 18.1

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

$2 x y^{\frac{5}{2}} + x^{\frac{3}{2}} y^{\frac{5}{2}} - 5 x^{\frac{7}{2}} y \cdot y^{3 (\frac{1}{2})}$

Step 18.2

Combine $3$ and $\frac{1}{2}$ .

$2 x y^{\frac{5}{2}} + x^{\frac{3}{2}} y^{\frac{5}{2}} - 5 x^{\frac{7}{2}} y \cdot y^{\frac{3}{2}}$

Step 19

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

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Step 19.1

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

$2 x y^{\frac{5}{2}} + x^{\frac{3}{2}} y^{\frac{5}{2}} - 5 x^{\frac{7}{2}} (y^{\frac{3}{2}} y)$

Step 19.2

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

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Step 19.2.1

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

$2 x y^{\frac{5}{2}} + x^{\frac{3}{2}} y^{\frac{5}{2}} - 5 x^{\frac{7}{2}} (y^{\frac{3}{2}} y^{1})$

Step 19.2.2

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

$2 x y^{\frac{5}{2}} + x^{\frac{3}{2}} y^{\frac{5}{2}} - 5 x^{\frac{7}{2}} y^{\frac{3}{2} + 1}$

Step 19.3

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

$2 x y^{\frac{5}{2}} + x^{\frac{3}{2}} y^{\frac{5}{2}} - 5 x^{\frac{7}{2}} y^{\frac{3}{2} + \frac{2}{2}}$

Step 19.4

Combine the numerators over the common denominator.

$2 x y^{\frac{5}{2}} + x^{\frac{3}{2}} y^{\frac{5}{2}} - 5 x^{\frac{7}{2}} y^{\frac{3 + 2}{2}}$

Step 19.5

Add $3$ and $2$ .

$2 x y^{\frac{5}{2}} + x^{\frac{3}{2}} y^{\frac{5}{2}} - 5 x^{\frac{7}{2}} y^{\frac{5}{2}}$

Step 20

Rewrite $2 x y^{\frac{5}{2}} + x^{\frac{3}{2}} y^{\frac{5}{2}} - 5 x^{\frac{7}{2}} y^{\frac{5}{2}}$ in a factored form.

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Step 20.1

Factor $x y^{\frac{5}{2}}$ out of $2 x y^{\frac{5}{2}} + x^{\frac{3}{2}} y^{\frac{5}{2}} - 5 x^{\frac{7}{2}} y^{\frac{5}{2}}$ .

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Step 20.1.1

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

$x y^{\frac{5}{2}} (2) + x^{\frac{3}{2}} y^{\frac{5}{2}} - 5 x^{\frac{7}{2}} y^{\frac{5}{2}}$

Step 20.1.2

Factor $x y^{\frac{5}{2}}$ out of $x^{\frac{3}{2}} y^{\frac{5}{2}}$ .

$x y^{\frac{5}{2}} (2) + x y^{\frac{5}{2}} (x^{\frac{1}{2}}) - 5 x^{\frac{7}{2}} y^{\frac{5}{2}}$

Step 20.1.3

Factor $x y^{\frac{5}{2}}$ out of $- 5 x^{\frac{7}{2}} y^{\frac{5}{2}}$ .

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

Step 20.1.4

Factor $x y^{\frac{5}{2}}$ out of $x y^{\frac{5}{2}} (2) + x y^{\frac{5}{2}} (x^{\frac{1}{2}})$ .

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

Step 20.1.5

Factor $x y^{\frac{5}{2}}$ out of $x y^{\frac{5}{2}} (2 + x^{\frac{1}{2}}) + x y^{\frac{5}{2}} (- 5 x^{\frac{5}{2}})$ .

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

Step 20.2

Reorder terms.

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