Linear Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

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

Step 1.2

Any root of $1$ is $1$ .

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

Step 1.3

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.4

Combine and simplify the denominator.

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Step 1.4.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.4.2

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

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

Step 1.4.3

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

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

Step 1.4.4

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

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

Step 1.4.5

Add $1$ and $1$ .

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

Step 1.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.4.6.1

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

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

Step 1.4.6.2

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

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

Step 1.4.6.3

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

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

Step 1.4.6.4

Cancel the common factor of $2$ .

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Step 1.4.6.4.1

Cancel the common factor.

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

Step 1.4.6.4.2

Rewrite the expression.

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

Step 1.4.6.5

Evaluate the exponent.

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

Step 1.5

Apply the product rule to $\frac{\sqrt{3}}{3}$ .

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

Step 1.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.6.1

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

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

Step 1.6.2

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

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

Step 1.6.3

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

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

Step 1.6.4

Cancel the common factor of $2$ .

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Step 1.6.4.1

Cancel the common factor.

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

Step 1.6.4.2

Rewrite the expression.

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

Step 1.6.5

Evaluate the exponent.

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

Step 1.7

Raise $3$ to the power of $2$ .

$\frac{3}{9} + {(\sqrt{\frac{2}{3}})}^{2} + \sqrt{{(\frac{1}{3})}^{2}}$

Step 1.8

Cancel the common factor of $3$ and $9$ .

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Step 1.8.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{9} + {(\sqrt{\frac{2}{3}})}^{2} + \sqrt{{(\frac{1}{3})}^{2}}$

Step 1.8.2

Cancel the common factors.

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Step 1.8.2.1

Factor $3$ out of $9$ .

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

Step 1.8.2.2

Cancel the common factor.

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

Step 1.8.2.3

Rewrite the expression.

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

Step 1.9

Rewrite $\sqrt{\frac{2}{3}}$ as $\frac{\sqrt{2}}{\sqrt{3}}$ .

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

Step 1.10

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.11

Combine and simplify the denominator.

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Step 1.11.1

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.11.2

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

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

Step 1.11.3

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

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

Step 1.11.4

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

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

Step 1.11.5

Add $1$ and $1$ .

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

Step 1.11.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.11.6.1

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

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

Step 1.11.6.2

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

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

Step 1.11.6.3

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

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

Step 1.11.6.4

Cancel the common factor of $2$ .

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Step 1.11.6.4.1

Cancel the common factor.

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

Step 1.11.6.4.2

Rewrite the expression.

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

Step 1.11.6.5

Evaluate the exponent.

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

Step 1.12

Simplify the numerator.

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Step 1.12.1

Combine using the product rule for radicals.

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

Step 1.12.2

Multiply $2$ by $3$ .

$\frac{1}{3} + {(\frac{\sqrt{6}}{3})}^{2} + \sqrt{{(\frac{1}{3})}^{2}}$

Step 1.13

Apply the product rule to $\frac{\sqrt{6}}{3}$ .

$\frac{1}{3} + \frac{{\sqrt{6}}^{2}}{3^{2}} + \sqrt{{(\frac{1}{3})}^{2}}$

Step 1.14

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 1.14.1

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

$\frac{1}{3} + \frac{{(6^{\frac{1}{2}})}^{2}}{3^{2}} + \sqrt{{(\frac{1}{3})}^{2}}$

Step 1.14.2

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

$\frac{1}{3} + \frac{6^{\frac{1}{2} \cdot 2}}{3^{2}} + \sqrt{{(\frac{1}{3})}^{2}}$

Step 1.14.3

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

$\frac{1}{3} + \frac{6^{\frac{2}{2}}}{3^{2}} + \sqrt{{(\frac{1}{3})}^{2}}$

Step 1.14.4

Cancel the common factor of $2$ .

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Step 1.14.4.1

Cancel the common factor.

$\frac{1}{3} + \frac{6^{\frac{2}{2}}}{3^{2}} + \sqrt{{(\frac{1}{3})}^{2}}$

Step 1.14.4.2

Rewrite the expression.

$\frac{1}{3} + \frac{6^{1}}{3^{2}} + \sqrt{{(\frac{1}{3})}^{2}}$

Step 1.14.5

Evaluate the exponent.

$\frac{1}{3} + \frac{6}{3^{2}} + \sqrt{{(\frac{1}{3})}^{2}}$

Step 1.15

Raise $3$ to the power of $2$ .

$\frac{1}{3} + \frac{6}{9} + \sqrt{{(\frac{1}{3})}^{2}}$

Step 1.16

Cancel the common factor of $6$ and $9$ .

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Step 1.16.1

Factor $3$ out of $6$ .

$\frac{1}{3} + \frac{3 (2)}{9} + \sqrt{{(\frac{1}{3})}^{2}}$

Step 1.16.2

Cancel the common factors.

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Step 1.16.2.1

Factor $3$ out of $9$ .

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

Step 1.16.2.2

Cancel the common factor.

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

Step 1.16.2.3

Rewrite the expression.

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

Step 1.17

Pull terms out from under the radical, assuming positive real numbers.

$\frac{1}{3} + \frac{2}{3} + \frac{1}{3}$

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{1 + 2 + 1}{3}$

Step 2.2

Simplify by adding numbers.

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Step 2.2.1

Add $1$ and $2$ .

$\frac{3 + 1}{3}$

Step 2.2.2

Add $3$ and $1$ .

$\frac{4}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{4}{3}$

Decimal Form:

$1.\overline{3}$

Mixed Number Form:

$1 \frac{1}{3}$