Finite Math Examples

$\sqrt{\frac{85 \cdot (597271.25) - 49456056.25}{85 (85 - 1)}}$

Step 1

Simplify the expression.

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Step 1.1

Multiply $85$ by $597271.25$ .

$\sqrt{\frac{50768056.25 - 49456056.25}{85 (85 - 1)}}$

Step 1.2

Subtract $49456056.25$ from $50768056.25$ .

$\sqrt{\frac{1312000}{85 (85 - 1)}}$

Step 1.3

Subtract $1$ from $85$ .

$\sqrt{\frac{1312000}{85 \cdot 84}}$

Step 2

Reduce the expression $\frac{1312000}{85 \cdot 84}$ by cancelling the common factors.

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Step 2.1

Factor $5$ out of $1312000$ .

$\sqrt{\frac{5 (262400)}{85 \cdot 84}}$

Step 2.2

Factor $5$ out of $85 \cdot 84$ .

$\sqrt{\frac{5 (262400)}{5 (17 \cdot 84)}}$

Step 2.3

Cancel the common factor.

$\sqrt{\frac{5 \cdot 262400}{5 (17 \cdot 84)}}$

Step 2.4

Rewrite the expression.

$\sqrt{\frac{262400}{17 \cdot 84}}$

Step 3

Cancel the common factor of $262400$ and $84$ .

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Step 3.1

Factor $4$ out of $262400$ .

$\sqrt{\frac{4 (65600)}{17 \cdot 84}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $4$ out of $17 \cdot 84$ .

$\sqrt{\frac{4 (65600)}{4 (17 \cdot 21)}}$

Step 3.2.2

Cancel the common factor.

$\sqrt{\frac{4 \cdot 65600}{4 (17 \cdot 21)}}$

Step 3.2.3

Rewrite the expression.

$\sqrt{\frac{65600}{17 \cdot 21}}$

Step 4

Multiply $17$ by $21$ .

$\sqrt{\frac{65600}{357}}$

Step 5

Rewrite $\sqrt{\frac{65600}{357}}$ as $\frac{\sqrt{65600}}{\sqrt{357}}$ .

$\frac{\sqrt{65600}}{\sqrt{357}}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite $65600$ as $40^{2} \cdot 41$ .

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Step 6.1.1

Factor $1600$ out of $65600$ .

$\frac{\sqrt{1600 (41)}}{\sqrt{357}}$

Step 6.1.2

Rewrite $1600$ as $40^{2}$ .

$\frac{\sqrt{40^{2} \cdot 41}}{\sqrt{357}}$

Step 6.2

Pull terms out from under the radical.

$\frac{40 \sqrt{41}}{\sqrt{357}}$

Step 7

Multiply $\frac{40 \sqrt{41}}{\sqrt{357}}$ by $\frac{\sqrt{357}}{\sqrt{357}}$ .

$\frac{40 \sqrt{41}}{\sqrt{357}} \cdot \frac{\sqrt{357}}{\sqrt{357}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{40 \sqrt{41}}{\sqrt{357}}$ by $\frac{\sqrt{357}}{\sqrt{357}}$ .

$\frac{40 \sqrt{41} \sqrt{357}}{\sqrt{357} \sqrt{357}}$

Step 8.2

Raise $\sqrt{357}$ to the power of $1$ .

$\frac{40 \sqrt{41} \sqrt{357}}{{\sqrt{357}}^{1} \sqrt{357}}$

Step 8.3

Raise $\sqrt{357}$ to the power of $1$ .

$\frac{40 \sqrt{41} \sqrt{357}}{{\sqrt{357}}^{1} {\sqrt{357}}^{1}}$

Step 8.4

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

$\frac{40 \sqrt{41} \sqrt{357}}{{\sqrt{357}}^{1 + 1}}$

Step 8.5

Add $1$ and $1$ .

$\frac{40 \sqrt{41} \sqrt{357}}{{\sqrt{357}}^{2}}$

Step 8.6

Rewrite ${\sqrt{357}}^{2}$ as $357$ .

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Step 8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{357}$ as $357^{\frac{1}{2}}$ .

$\frac{40 \sqrt{41} \sqrt{357}}{{(357^{\frac{1}{2}})}^{2}}$

Step 8.6.2

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

$\frac{40 \sqrt{41} \sqrt{357}}{357^{\frac{1}{2} \cdot 2}}$

Step 8.6.3

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

$\frac{40 \sqrt{41} \sqrt{357}}{357^{\frac{2}{2}}}$

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

$\frac{40 \sqrt{41} \sqrt{357}}{357^{\frac{2}{2}}}$

Step 8.6.4.2

Rewrite the expression.

$\frac{40 \sqrt{41} \sqrt{357}}{357^{1}}$

Step 8.6.5

Evaluate the exponent.

$\frac{40 \sqrt{41} \sqrt{357}}{357}$

Step 9

Simplify the numerator.

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Step 9.1

Combine using the product rule for radicals.

$\frac{40 \sqrt{357 \cdot 41}}{357}$

Step 9.2

Multiply $357$ by $41$ .

$\frac{40 \sqrt{14637}}{357}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{40 \sqrt{14637}}{357}$

Decimal Form:

$13.55557086 \dots$