Finite Math Examples

$\sqrt{\frac{{(- 8)}^{2} \cdot 16 + {(- 3)}^{2} \cdot 42 + {(2)}^{2} \cdot 22 + {(7)}^{2} \cdot 13 + {(12)}^{2} \cdot 2 + {(7)}^{2} \cdot 3 + {(32 - 10)}^{2} \cdot 2}{99}}$

Step 1

Raise $- 8$ to the power of $2$ .

$\sqrt{\frac{64 \cdot 16 + {(- 3)}^{2} \cdot 42 + {(2)}^{2} \cdot 22 + {(7)}^{2} \cdot 13 + {(12)}^{2} \cdot 2 + {(7)}^{2} \cdot 3 + {(32 - 10)}^{2} \cdot 2}{99}}$

Step 2

Multiply $64$ by $16$ .

$\sqrt{\frac{1024 + {(- 3)}^{2} \cdot 42 + {(2)}^{2} \cdot 22 + {(7)}^{2} \cdot 13 + {(12)}^{2} \cdot 2 + {(7)}^{2} \cdot 3 + {(32 - 10)}^{2} \cdot 2}{99}}$

Step 3

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

$\sqrt{\frac{1024 + 9 \cdot 42 + {(2)}^{2} \cdot 22 + {(7)}^{2} \cdot 13 + {(12)}^{2} \cdot 2 + {(7)}^{2} \cdot 3 + {(32 - 10)}^{2} \cdot 2}{99}}$

Step 4

Multiply $9$ by $42$ .

$\sqrt{\frac{1024 + 378 + {(2)}^{2} \cdot 22 + {(7)}^{2} \cdot 13 + {(12)}^{2} \cdot 2 + {(7)}^{2} \cdot 3 + {(32 - 10)}^{2} \cdot 2}{99}}$

Step 5

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

$\sqrt{\frac{1024 + 378 + 4 \cdot 22 + {(7)}^{2} \cdot 13 + {(12)}^{2} \cdot 2 + {(7)}^{2} \cdot 3 + {(32 - 10)}^{2} \cdot 2}{99}}$

Step 6

Multiply $4$ by $22$ .

$\sqrt{\frac{1024 + 378 + 88 + {(7)}^{2} \cdot 13 + {(12)}^{2} \cdot 2 + {(7)}^{2} \cdot 3 + {(32 - 10)}^{2} \cdot 2}{99}}$

Step 7

Raise $7$ to the power of $2$ .

$\sqrt{\frac{1024 + 378 + 88 + 49 \cdot 13 + {(12)}^{2} \cdot 2 + {(7)}^{2} \cdot 3 + {(32 - 10)}^{2} \cdot 2}{99}}$

Step 8

Multiply $49$ by $13$ .

$\sqrt{\frac{1024 + 378 + 88 + 637 + {(12)}^{2} \cdot 2 + {(7)}^{2} \cdot 3 + {(32 - 10)}^{2} \cdot 2}{99}}$

Step 9

Raise $12$ to the power of $2$ .

$\sqrt{\frac{1024 + 378 + 88 + 637 + 144 \cdot 2 + {(7)}^{2} \cdot 3 + {(32 - 10)}^{2} \cdot 2}{99}}$

Step 10

Multiply $144$ by $2$ .

$\sqrt{\frac{1024 + 378 + 88 + 637 + 288 + {(7)}^{2} \cdot 3 + {(32 - 10)}^{2} \cdot 2}{99}}$

Step 11

Raise $7$ to the power of $2$ .

$\sqrt{\frac{1024 + 378 + 88 + 637 + 288 + 49 \cdot 3 + {(32 - 10)}^{2} \cdot 2}{99}}$

Step 12

Multiply $49$ by $3$ .

$\sqrt{\frac{1024 + 378 + 88 + 637 + 288 + 147 + {(32 - 10)}^{2} \cdot 2}{99}}$

Step 13

Subtract $10$ from $32$ .

$\sqrt{\frac{1024 + 378 + 88 + 637 + 288 + 147 + 22^{2} \cdot 2}{99}}$

Step 14

Raise $22$ to the power of $2$ .

$\sqrt{\frac{1024 + 378 + 88 + 637 + 288 + 147 + 484 \cdot 2}{99}}$

Step 15

Multiply $484$ by $2$ .

$\sqrt{\frac{1024 + 378 + 88 + 637 + 288 + 147 + 968}{99}}$

Step 16

Add $1024$ and $378$ .

$\sqrt{\frac{1402 + 88 + 637 + 288 + 147 + 968}{99}}$

Step 17

Add $1402$ and $88$ .

$\sqrt{\frac{1490 + 637 + 288 + 147 + 968}{99}}$

Step 18

Add $1490$ and $637$ .

$\sqrt{\frac{2127 + 288 + 147 + 968}{99}}$

Step 19

Add $2127$ and $288$ .

$\sqrt{\frac{2415 + 147 + 968}{99}}$

Step 20

Add $2415$ and $147$ .

$\sqrt{\frac{2562 + 968}{99}}$

Step 21

Add $2562$ and $968$ .

$\sqrt{\frac{3530}{99}}$

Step 22

Rewrite $\sqrt{\frac{3530}{99}}$ as $\frac{\sqrt{3530}}{\sqrt{99}}$ .

$\frac{\sqrt{3530}}{\sqrt{99}}$

Step 23

Simplify the denominator.

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Step 23.1

Rewrite $99$ as $3^{2} \cdot 11$ .

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Step 23.1.1

Factor $9$ out of $99$ .

$\frac{\sqrt{3530}}{\sqrt{9 (11)}}$

Step 23.1.2

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

$\frac{\sqrt{3530}}{\sqrt{3^{2} \cdot 11}}$

Step 23.2

Pull terms out from under the radical.

$\frac{\sqrt{3530}}{3 \sqrt{11}}$

Step 24

Multiply $\frac{\sqrt{3530}}{3 \sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{\sqrt{3530}}{3 \sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}}$

Step 25

Combine and simplify the denominator.

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Step 25.1

Multiply $\frac{\sqrt{3530}}{3 \sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{\sqrt{3530} \sqrt{11}}{3 \sqrt{11} \sqrt{11}}$

Step 25.2

Move $\sqrt{11}$ .

$\frac{\sqrt{3530} \sqrt{11}}{3 (\sqrt{11} \sqrt{11})}$

Step 25.3

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{\sqrt{3530} \sqrt{11}}{3 ({\sqrt{11}}^{1} \sqrt{11})}$

Step 25.4

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{\sqrt{3530} \sqrt{11}}{3 ({\sqrt{11}}^{1} {\sqrt{11}}^{1})}$

Step 25.5

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

$\frac{\sqrt{3530} \sqrt{11}}{3 {\sqrt{11}}^{1 + 1}}$

Step 25.6

Add $1$ and $1$ .

$\frac{\sqrt{3530} \sqrt{11}}{3 {\sqrt{11}}^{2}}$

Step 25.7

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 25.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\frac{\sqrt{3530} \sqrt{11}}{3 {(11^{\frac{1}{2}})}^{2}}$

Step 25.7.2

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

$\frac{\sqrt{3530} \sqrt{11}}{3 \cdot 11^{\frac{1}{2} \cdot 2}}$

Step 25.7.3

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

$\frac{\sqrt{3530} \sqrt{11}}{3 \cdot 11^{\frac{2}{2}}}$

Step 25.7.4

Cancel the common factor of $2$ .

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Step 25.7.4.1

Cancel the common factor.

$\frac{\sqrt{3530} \sqrt{11}}{3 \cdot 11^{\frac{2}{2}}}$

Step 25.7.4.2

Rewrite the expression.

$\frac{\sqrt{3530} \sqrt{11}}{3 \cdot 11^{1}}$

Step 25.7.5

Evaluate the exponent.

$\frac{\sqrt{3530} \sqrt{11}}{3 \cdot 11}$

Step 26

Simplify the numerator.

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Step 26.1

Combine using the product rule for radicals.

$\frac{\sqrt{3530 \cdot 11}}{3 \cdot 11}$

Step 26.2

Multiply $3530$ by $11$ .

$\frac{\sqrt{38830}}{3 \cdot 11}$

Step 27

Multiply $3$ by $11$ .

$\frac{\sqrt{38830}}{33}$

Step 28

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{38830}}{33}$

Decimal Form:

$5.97131188 \dots$