Algebra Examples

$\frac{10 \sqrt{2 x^{20}} + 7 \sqrt{8 x^{20}}}{\sqrt{18 x^{20}}}$

Step 1

Simplify the numerator.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x^{20}}$ as ${(2 x^{20})}^{\frac{1}{2}}$ .

$\frac{10 {(2 x^{20})}^{\frac{1}{2}} + 7 \sqrt{8 x^{20}}}{\sqrt{18 x^{20}}}$

Step 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 x^{20}}$ as ${(8 x^{20})}^{\frac{1}{2}}$ .

$\frac{10 {(2 x^{20})}^{\frac{1}{2}} + 7 {(8 x^{20})}^{\frac{1}{2}}}{\sqrt{18 x^{20}}}$

Step 1.3

Apply the product rule to $2 x^{20}$ .

$\frac{10 (2^{\frac{1}{2}} {(x^{20})}^{\frac{1}{2}}) + 7 {(8 x^{20})}^{\frac{1}{2}}}{\sqrt{18 x^{20}}}$

Step 1.4

Multiply the exponents in ${(x^{20})}^{\frac{1}{2}}$ .

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Step 1.4.1

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

$\frac{10 (2^{\frac{1}{2}} x^{20 (\frac{1}{2})}) + 7 {(8 x^{20})}^{\frac{1}{2}}}{\sqrt{18 x^{20}}}$

Step 1.4.2

Cancel the common factor of $2$ .

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Step 1.4.2.1

Factor $2$ out of $20$ .

$\frac{10 (2^{\frac{1}{2}} x^{2 (10) \frac{1}{2}}) + 7 {(8 x^{20})}^{\frac{1}{2}}}{\sqrt{18 x^{20}}}$

Step 1.4.2.2

Cancel the common factor.

$\frac{10 (2^{\frac{1}{2}} x^{2 \cdot 10 \frac{1}{2}}) + 7 {(8 x^{20})}^{\frac{1}{2}}}{\sqrt{18 x^{20}}}$

Step 1.4.2.3

Rewrite the expression.

$\frac{10 (2^{\frac{1}{2}} x^{10}) + 7 {(8 x^{20})}^{\frac{1}{2}}}{\sqrt{18 x^{20}}}$

Step 1.5

Apply the product rule to $8 x^{20}$ .

$\frac{10 \cdot 2^{\frac{1}{2}} x^{10} + 7 (8^{\frac{1}{2}} {(x^{20})}^{\frac{1}{2}})}{\sqrt{18 x^{20}}}$

Step 1.6

Multiply the exponents in ${(x^{20})}^{\frac{1}{2}}$ .

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Step 1.6.1

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

$\frac{10 \cdot 2^{\frac{1}{2}} x^{10} + 7 (8^{\frac{1}{2}} x^{20 (\frac{1}{2})})}{\sqrt{18 x^{20}}}$

Step 1.6.2

Cancel the common factor of $2$ .

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Step 1.6.2.1

Factor $2$ out of $20$ .

$\frac{10 \cdot 2^{\frac{1}{2}} x^{10} + 7 (8^{\frac{1}{2}} x^{2 (10) \frac{1}{2}})}{\sqrt{18 x^{20}}}$

Step 1.6.2.2

Cancel the common factor.

$\frac{10 \cdot 2^{\frac{1}{2}} x^{10} + 7 (8^{\frac{1}{2}} x^{2 \cdot 10 \frac{1}{2}})}{\sqrt{18 x^{20}}}$

Step 1.6.2.3

Rewrite the expression.

$\frac{10 \cdot 2^{\frac{1}{2}} x^{10} + 7 (8^{\frac{1}{2}} x^{10})}{\sqrt{18 x^{20}}}$

Step 1.7

Factor $x^{10}$ out of $10 \cdot 2^{\frac{1}{2}} x^{10} + 7 \cdot 8^{\frac{1}{2}} x^{10}$ .

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Step 1.7.1

Factor $x^{10}$ out of $10 \cdot 2^{\frac{1}{2}} x^{10}$ .

$\frac{x^{10} (10 \cdot 2^{\frac{1}{2}}) + 7 \cdot 8^{\frac{1}{2}} x^{10}}{\sqrt{18 x^{20}}}$

Step 1.7.2

Factor $x^{10}$ out of $7 \cdot 8^{\frac{1}{2}} x^{10}$ .

$\frac{x^{10} (10 \cdot 2^{\frac{1}{2}}) + x^{10} (7 \cdot 8^{\frac{1}{2}})}{\sqrt{18 x^{20}}}$

Step 1.7.3

Factor $x^{10}$ out of $x^{10} (10 \cdot 2^{\frac{1}{2}}) + x^{10} (7 \cdot 8^{\frac{1}{2}})$ .

$\frac{x^{10} (10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}})}{\sqrt{18 x^{20}}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $18 x^{20}$ as ${(3 x^{10})}^{2} \cdot 2$ .

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Step 2.1.1

Factor $9$ out of $18$ .

$\frac{x^{10} (10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}})}{\sqrt{9 (2) x^{20}}}$

Step 2.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{x^{10} (10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}})}{\sqrt{3^{2} \cdot 2 x^{20}}}$

Step 2.1.3

Rewrite $x^{20}$ as ${(x^{10})}^{2}$ .

$\frac{x^{10} (10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}})}{\sqrt{3^{2} \cdot 2 {(x^{10})}^{2}}}$

Step 2.1.4

Move $2$ .

$\frac{x^{10} (10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}})}{\sqrt{3^{2} {(x^{10})}^{2} \cdot 2}}$

Step 2.1.5

Rewrite $3^{2} {(x^{10})}^{2}$ as ${(3 x^{10})}^{2}$ .

$\frac{x^{10} (10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}})}{\sqrt{{(3 x^{10})}^{2} \cdot 2}}$

Step 2.2

Pull terms out from under the radical.

$\frac{x^{10} (10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}})}{3 x^{10} \sqrt{2}}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor.

$\frac{x^{10} (10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}})}{3 x^{10} \sqrt{2}}$

Step 3.2

Rewrite the expression.

$\frac{10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}}}{3 \sqrt{2}}$

Step 4

Multiply $\frac{10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}}}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}}}{3 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}}}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{(10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}}) \sqrt{2}}{3 \sqrt{2} \sqrt{2}}$

Step 5.2

Move $\sqrt{2}$ .

$\frac{(10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}}) \sqrt{2}}{3 (\sqrt{2} \sqrt{2})}$

Step 5.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{(10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}}) \sqrt{2}}{3 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 5.4

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{(10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}}) \sqrt{2}}{3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 5.5

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

$\frac{(10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}}) \sqrt{2}}{3 {\sqrt{2}}^{1 + 1}}$

Step 5.6

Add $1$ and $1$ .

$\frac{(10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}}) \sqrt{2}}{3 {\sqrt{2}}^{2}}$

Step 5.7

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

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Step 5.7.1

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

$\frac{(10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}}) \sqrt{2}}{3 {(2^{\frac{1}{2}})}^{2}}$

Step 5.7.2

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

$\frac{(10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}}) \sqrt{2}}{3 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 5.7.3

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

$\frac{(10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}}) \sqrt{2}}{3 \cdot 2^{\frac{2}{2}}}$

Step 5.7.4

Cancel the common factor of $2$ .

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Step 5.7.4.1

Cancel the common factor.

$\frac{(10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}}) \sqrt{2}}{3 \cdot 2^{\frac{2}{2}}}$

Step 5.7.4.2

Rewrite the expression.

$\frac{(10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}}) \sqrt{2}}{3 \cdot 2^{1}}$

Step 5.7.5

Evaluate the exponent.

$\frac{(10 \cdot 2^{\frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}}) \sqrt{2}}{3 \cdot 2}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{10 \cdot 2^{\frac{1}{2}} \sqrt{2} + 7 \cdot 8^{\frac{1}{2}} \sqrt{2}}{3 \cdot 2}$

Step 6.2

Multiply $10 \cdot 2^{\frac{1}{2}} \sqrt{2}$ .

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Step 6.2.1

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

$\frac{10 \cdot (2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}}) + 7 \cdot 8^{\frac{1}{2}} \sqrt{2}}{3 \cdot 2}$

Step 6.2.2

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

$\frac{10 \cdot 2^{\frac{1}{2} + \frac{1}{2}} + 7 \cdot 8^{\frac{1}{2}} \sqrt{2}}{3 \cdot 2}$

Step 6.2.3

Combine the numerators over the common denominator.

$\frac{10 \cdot 2^{\frac{1 + 1}{2}} + 7 \cdot 8^{\frac{1}{2}} \sqrt{2}}{3 \cdot 2}$

Step 6.2.4

Add $1$ and $1$ .

$\frac{10 \cdot 2^{\frac{2}{2}} + 7 \cdot 8^{\frac{1}{2}} \sqrt{2}}{3 \cdot 2}$

Step 6.2.5

Cancel the common factor of $2$ .

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Step 6.2.5.1

Cancel the common factor.

$\frac{10 \cdot 2^{\frac{2}{2}} + 7 \cdot 8^{\frac{1}{2}} \sqrt{2}}{3 \cdot 2}$

Step 6.2.5.2

Rewrite the expression.

$\frac{10 \cdot 2^{1} + 7 \cdot 8^{\frac{1}{2}} \sqrt{2}}{3 \cdot 2}$

Step 6.3

Multiply $7 \cdot 8^{\frac{1}{2}} \sqrt{2}$ .

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Step 6.3.1

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

$\frac{10 \cdot 2^{1} + 7 \cdot (2^{\frac{1}{2}} \cdot 8^{\frac{1}{2}})}{3 \cdot 2}$

Step 6.3.2

Rewrite $8$ as $2^{3}$ .

$\frac{10 \cdot 2^{1} + 7 \cdot (2^{\frac{1}{2}} \cdot {(2^{3})}^{\frac{1}{2}})}{3 \cdot 2}$

Step 6.3.3

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

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Step 6.3.3.1

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

$\frac{10 \cdot 2^{1} + 7 \cdot (2^{\frac{1}{2}} \cdot 2^{3 (\frac{1}{2})})}{3 \cdot 2}$

Step 6.3.3.2

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

$\frac{10 \cdot 2^{1} + 7 \cdot (2^{\frac{1}{2}} \cdot 2^{\frac{3}{2}})}{3 \cdot 2}$

Step 6.3.4

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

$\frac{10 \cdot 2^{1} + 7 \cdot 2^{\frac{1}{2} + \frac{3}{2}}}{3 \cdot 2}$

Step 6.3.5

Combine the numerators over the common denominator.

$\frac{10 \cdot 2^{1} + 7 \cdot 2^{\frac{1 + 3}{2}}}{3 \cdot 2}$

Step 6.3.6

Add $1$ and $3$ .

$\frac{10 \cdot 2^{1} + 7 \cdot 2^{\frac{4}{2}}}{3 \cdot 2}$

Step 6.3.7

Cancel the common factor of $4$ and $2$ .

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Step 6.3.7.1

Factor $2$ out of $4$ .

$\frac{10 \cdot 2^{1} + 7 \cdot 2^{\frac{2 \cdot 2}{2}}}{3 \cdot 2}$

Step 6.3.7.2

Cancel the common factors.

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Step 6.3.7.2.1

Factor $2$ out of $2$ .

$\frac{10 \cdot 2^{1} + 7 \cdot 2^{\frac{2 \cdot 2}{2 (1)}}}{3 \cdot 2}$

Step 6.3.7.2.2

Cancel the common factor.

$\frac{10 \cdot 2^{1} + 7 \cdot 2^{\frac{2 \cdot 2}{2 \cdot 1}}}{3 \cdot 2}$

Step 6.3.7.2.3

Rewrite the expression.

$\frac{10 \cdot 2^{1} + 7 \cdot 2^{\frac{2}{1}}}{3 \cdot 2}$

Step 6.3.7.2.4

Divide $2$ by $1$ .

$\frac{10 \cdot 2^{1} + 7 \cdot 2^{2}}{3 \cdot 2}$

Step 6.4

Evaluate the exponent.

$\frac{10 \cdot 2 + 7 \cdot 2^{2}}{3 \cdot 2}$

Step 6.5

Multiply $10$ by $2$ .

$\frac{20 + 7 \cdot 2^{2}}{3 \cdot 2}$

Step 6.6

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

$\frac{20 + 7 \cdot 4}{3 \cdot 2}$

Step 6.7

Multiply $7$ by $4$ .

$\frac{20 + 28}{3 \cdot 2}$

Step 6.8

Add $20$ and $28$ .

$\frac{48}{3 \cdot 2}$

Step 7

Simplify the expression.

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Step 7.1

Multiply $3$ by $2$ .

$\frac{48}{6}$

Step 7.2

Divide $48$ by $6$ .

$8$