Linear Algebra Examples

$- \frac{\sqrt{256}}{\sqrt{545}} + \frac{\sqrt{45}}{\sqrt{109}} + \frac{\sqrt{1024}}{\sqrt{545}}$

Step 1

Combine the numerators over the common denominator.

$\frac{- \sqrt{256} + \sqrt{1024}}{\sqrt{545}} + \frac{\sqrt{45}}{\sqrt{109}}$

Step 2

Simplify each term.

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Step 2.1

Rewrite $256$ as $16^{2}$ .

$\frac{- \sqrt{16^{2}} + \sqrt{1024}}{\sqrt{545}} + \frac{\sqrt{45}}{\sqrt{109}}$

Step 2.2

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

$\frac{- 1 \cdot 16 + \sqrt{1024}}{\sqrt{545}} + \frac{\sqrt{45}}{\sqrt{109}}$

Step 2.3

Multiply $- 1$ by $16$ .

$\frac{- 16 + \sqrt{1024}}{\sqrt{545}} + \frac{\sqrt{45}}{\sqrt{109}}$

Step 2.4

Rewrite $1024$ as $32^{2}$ .

$\frac{- 16 + \sqrt{32^{2}}}{\sqrt{545}} + \frac{\sqrt{45}}{\sqrt{109}}$

Step 2.5

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

$\frac{- 16 + 32}{\sqrt{545}} + \frac{\sqrt{45}}{\sqrt{109}}$

Step 3

Add $- 16$ and $32$ .

$\frac{16}{\sqrt{545}} + \frac{\sqrt{45}}{\sqrt{109}}$

Step 4

Simplify each term.

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Step 4.1

Multiply $\frac{16}{\sqrt{545}}$ by $\frac{\sqrt{545}}{\sqrt{545}}$ .

$\frac{16}{\sqrt{545}} \cdot \frac{\sqrt{545}}{\sqrt{545}} + \frac{\sqrt{45}}{\sqrt{109}}$

Step 4.2

Combine and simplify the denominator.

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Step 4.2.1

Multiply $\frac{16}{\sqrt{545}}$ by $\frac{\sqrt{545}}{\sqrt{545}}$ .

$\frac{16 \sqrt{545}}{\sqrt{545} \sqrt{545}} + \frac{\sqrt{45}}{\sqrt{109}}$

Step 4.2.2

Raise $\sqrt{545}$ to the power of $1$ .

$\frac{16 \sqrt{545}}{{\sqrt{545}}^{1} \sqrt{545}} + \frac{\sqrt{45}}{\sqrt{109}}$

Step 4.2.3

Raise $\sqrt{545}$ to the power of $1$ .

$\frac{16 \sqrt{545}}{{\sqrt{545}}^{1} {\sqrt{545}}^{1}} + \frac{\sqrt{45}}{\sqrt{109}}$

Step 4.2.4

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

$\frac{16 \sqrt{545}}{{\sqrt{545}}^{1 + 1}} + \frac{\sqrt{45}}{\sqrt{109}}$

Step 4.2.5

Add $1$ and $1$ .

$\frac{16 \sqrt{545}}{{\sqrt{545}}^{2}} + \frac{\sqrt{45}}{\sqrt{109}}$

Step 4.2.6

Rewrite ${\sqrt{545}}^{2}$ as $545$ .

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Step 4.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{545}$ as $545^{\frac{1}{2}}$ .

$\frac{16 \sqrt{545}}{{(545^{\frac{1}{2}})}^{2}} + \frac{\sqrt{45}}{\sqrt{109}}$

Step 4.2.6.2

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

$\frac{16 \sqrt{545}}{545^{\frac{1}{2} \cdot 2}} + \frac{\sqrt{45}}{\sqrt{109}}$

Step 4.2.6.3

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

$\frac{16 \sqrt{545}}{545^{\frac{2}{2}}} + \frac{\sqrt{45}}{\sqrt{109}}$

Step 4.2.6.4

Cancel the common factor of $2$ .

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Step 4.2.6.4.1

Cancel the common factor.

$\frac{16 \sqrt{545}}{545^{\frac{2}{2}}} + \frac{\sqrt{45}}{\sqrt{109}}$

Step 4.2.6.4.2

Rewrite the expression.

$\frac{16 \sqrt{545}}{545^{1}} + \frac{\sqrt{45}}{\sqrt{109}}$

Step 4.2.6.5

Evaluate the exponent.

$\frac{16 \sqrt{545}}{545} + \frac{\sqrt{45}}{\sqrt{109}}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 4.3.1.1

Factor $9$ out of $45$ .

$\frac{16 \sqrt{545}}{545} + \frac{\sqrt{9 (5)}}{\sqrt{109}}$

Step 4.3.1.2

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

$\frac{16 \sqrt{545}}{545} + \frac{\sqrt{3^{2} \cdot 5}}{\sqrt{109}}$

Step 4.3.2

Pull terms out from under the radical.

$\frac{16 \sqrt{545}}{545} + \frac{3 \sqrt{5}}{\sqrt{109}}$

Step 4.4

Multiply $\frac{3 \sqrt{5}}{\sqrt{109}}$ by $\frac{\sqrt{109}}{\sqrt{109}}$ .

$\frac{16 \sqrt{545}}{545} + \frac{3 \sqrt{5}}{\sqrt{109}} \cdot \frac{\sqrt{109}}{\sqrt{109}}$

Step 4.5

Combine and simplify the denominator.

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Step 4.5.1

Multiply $\frac{3 \sqrt{5}}{\sqrt{109}}$ by $\frac{\sqrt{109}}{\sqrt{109}}$ .

$\frac{16 \sqrt{545}}{545} + \frac{3 \sqrt{5} \sqrt{109}}{\sqrt{109} \sqrt{109}}$

Step 4.5.2

Raise $\sqrt{109}$ to the power of $1$ .

$\frac{16 \sqrt{545}}{545} + \frac{3 \sqrt{5} \sqrt{109}}{{\sqrt{109}}^{1} \sqrt{109}}$

Step 4.5.3

Raise $\sqrt{109}$ to the power of $1$ .

$\frac{16 \sqrt{545}}{545} + \frac{3 \sqrt{5} \sqrt{109}}{{\sqrt{109}}^{1} {\sqrt{109}}^{1}}$

Step 4.5.4

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

$\frac{16 \sqrt{545}}{545} + \frac{3 \sqrt{5} \sqrt{109}}{{\sqrt{109}}^{1 + 1}}$

Step 4.5.5

Add $1$ and $1$ .

$\frac{16 \sqrt{545}}{545} + \frac{3 \sqrt{5} \sqrt{109}}{{\sqrt{109}}^{2}}$

Step 4.5.6

Rewrite ${\sqrt{109}}^{2}$ as $109$ .

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Step 4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{109}$ as $109^{\frac{1}{2}}$ .

$\frac{16 \sqrt{545}}{545} + \frac{3 \sqrt{5} \sqrt{109}}{{(109^{\frac{1}{2}})}^{2}}$

Step 4.5.6.2

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

$\frac{16 \sqrt{545}}{545} + \frac{3 \sqrt{5} \sqrt{109}}{109^{\frac{1}{2} \cdot 2}}$

Step 4.5.6.3

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

$\frac{16 \sqrt{545}}{545} + \frac{3 \sqrt{5} \sqrt{109}}{109^{\frac{2}{2}}}$

Step 4.5.6.4

Cancel the common factor of $2$ .

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Step 4.5.6.4.1

Cancel the common factor.

$\frac{16 \sqrt{545}}{545} + \frac{3 \sqrt{5} \sqrt{109}}{109^{\frac{2}{2}}}$

Step 4.5.6.4.2

Rewrite the expression.

$\frac{16 \sqrt{545}}{545} + \frac{3 \sqrt{5} \sqrt{109}}{109^{1}}$

Step 4.5.6.5

Evaluate the exponent.

$\frac{16 \sqrt{545}}{545} + \frac{3 \sqrt{5} \sqrt{109}}{109}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Combine using the product rule for radicals.

$\frac{16 \sqrt{545}}{545} + \frac{3 \sqrt{109 \cdot 5}}{109}$

Step 4.6.2

Multiply $109$ by $5$ .

$\frac{16 \sqrt{545}}{545} + \frac{3 \sqrt{545}}{109}$

Step 5

To write $\frac{3 \sqrt{545}}{109}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{16 \sqrt{545}}{545} + \frac{3 \sqrt{545}}{109} \cdot \frac{5}{5}$

Step 6

Write each expression with a common denominator of $545$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1

Multiply $\frac{3 \sqrt{545}}{109}$ by $\frac{5}{5}$ .

$\frac{16 \sqrt{545}}{545} + \frac{3 \sqrt{545} \cdot 5}{109 \cdot 5}$

Step 6.2

Multiply $109$ by $5$ .

$\frac{16 \sqrt{545}}{545} + \frac{3 \sqrt{545} \cdot 5}{545}$

Step 7

Combine the numerators over the common denominator.

$\frac{16 \sqrt{545} + 3 \sqrt{545} \cdot 5}{545}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $5$ by $3$ .

$\frac{16 \sqrt{545} + 15 \sqrt{545}}{545}$

Step 8.2

Add $16 \sqrt{545}$ and $15 \sqrt{545}$ .

$\frac{31 \sqrt{545}}{545}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{31 \sqrt{545}}{545}$

Decimal Form:

$1.32789410 \dots$