Basic Math Examples

(5285 - 5245) + 1.96 \sqrt{\frac{155^{2}}{395} + \frac{210^{2}}{395}}

$(5285 - 5245) + 1.96 \sqrt{\frac{155^{2}}{395} + \frac{210^{2}}{395}}$

Step 1

Simplify each term.

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Step 1.1

Combine the numerators over the common denominator.

5285 - 5245 + 1.96 \sqrt{\frac{155^{2} + 210^{2}}{395}}

$5285 - 5245 + 1.96 \sqrt{\frac{155^{2} + 210^{2}}{395}}$

Step 1.2

Simplify the numerator.

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Step 1.2.1

Raise

155

$155$ to the power of

2

$2$ .

5285 - 5245 + 1.96 \sqrt{\frac{24025 + 210^{2}}{395}}

$5285 - 5245 + 1.96 \sqrt{\frac{24025 + 210^{2}}{395}}$

Step 1.2.2

Raise

210

$210$ to the power of

2

$2$ .

5285 - 5245 + 1.96 \sqrt{\frac{24025 + 44100}{395}}

$5285 - 5245 + 1.96 \sqrt{\frac{24025 + 44100}{395}}$

Step 1.2.3

Add

24025

$24025$ and

44100

$44100$ .

5285 - 5245 + 1.96 \sqrt{\frac{68125}{395}}

$5285 - 5245 + 1.96 \sqrt{\frac{68125}{395}}$

5285 - 5245 + 1.96 \sqrt{\frac{68125}{395}}

$5285 - 5245 + 1.96 \sqrt{\frac{68125}{395}}$

Step 1.3

Cancel the common factor of

68125

$68125$ and

395

$395$ .

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Step 1.3.1

Factor

5

$5$ out of

68125

$68125$ .

5285 - 5245 + 1.96 \sqrt{\frac{5 (13625)}{395}}

$5285 - 5245 + 1.96 \sqrt{\frac{5 (13625)}{395}}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor

5

$5$ out of

395

$395$ .

5285 - 5245 + 1.96 \sqrt{\frac{5 \cdot 13625}{5 \cdot 79}}

$5285 - 5245 + 1.96 \sqrt{\frac{5 \cdot 13625}{5 \cdot 79}}$

Step 1.3.2.2

Cancel the common factor.

$5285 - 5245 + 1.96 \sqrt{\frac{5 \cdot 13625}{5 \cdot 79}}$

Step 1.3.2.3

Rewrite the expression.

$5285 - 5245 + 1.96 \sqrt{\frac{13625}{79}}$

Step 1.4

Rewrite

$\sqrt{\frac{13625}{79}}$ as

$\frac{\sqrt{13625}}{\sqrt{79}}$ .

$5285 - 5245 + 1.96 \frac{\sqrt{13625}}{\sqrt{79}}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Rewrite

$13625$ as

$5^{2} \cdot 545$ .

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Step 1.5.1.1

Factor

$25$ out of

$13625$ .

$5285 - 5245 + 1.96 \frac{\sqrt{25 (545)}}{\sqrt{79}}$

Step 1.5.1.2

Rewrite

$25$ as

$5^{2}$ .

$5285 - 5245 + 1.96 \frac{\sqrt{5^{2} \cdot 545}}{\sqrt{79}}$

Step 1.5.2

Pull terms out from under the radical.

$5285 - 5245 + 1.96 \frac{5 \sqrt{545}}{\sqrt{79}}$

Step 1.6

Multiply

$\frac{5 \sqrt{545}}{\sqrt{79}}$ by

$\frac{\sqrt{79}}{\sqrt{79}}$ .

$5285 - 5245 + 1.96 (\frac{5 \sqrt{545}}{\sqrt{79}} \cdot \frac{\sqrt{79}}{\sqrt{79}})$

Step 1.7

Combine and simplify the denominator.

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Step 1.7.1

Multiply

$\frac{5 \sqrt{545}}{\sqrt{79}}$ by

$\frac{\sqrt{79}}{\sqrt{79}}$ .

$5285 - 5245 + 1.96 \frac{5 \sqrt{545} \sqrt{79}}{\sqrt{79} \sqrt{79}}$

Step 1.7.2

Raise

$\sqrt{79}$ to the power of

$1$ .

$5285 - 5245 + 1.96 \frac{5 \sqrt{545} \sqrt{79}}{{\sqrt{79}}^{1} \sqrt{79}}$

Step 1.7.3

Raise

$\sqrt{79}$ to the power of

$1$ .

$5285 - 5245 + 1.96 \frac{5 \sqrt{545} \sqrt{79}}{{\sqrt{79}}^{1} {\sqrt{79}}^{1}}$

Step 1.7.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5285 - 5245 + 1.96 \frac{5 \sqrt{545} \sqrt{79}}{{\sqrt{79}}^{1 + 1}}$

Step 1.7.5

Add

$1$ and

$1$ .

$5285 - 5245 + 1.96 \frac{5 \sqrt{545} \sqrt{79}}{{\sqrt{79}}^{2}}$

Step 1.7.6

Rewrite

${\sqrt{79}}^{2}$ as

$79$ .

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Step 1.7.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{79}$ as

$79^{\frac{1}{2}}$ .

$5285 - 5245 + 1.96 \frac{5 \sqrt{545} \sqrt{79}}{{(79^{\frac{1}{2}})}^{2}}$

Step 1.7.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$5285 - 5245 + 1.96 \frac{5 \sqrt{545} \sqrt{79}}{79^{\frac{1}{2} \cdot 2}}$

Step 1.7.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$5285 - 5245 + 1.96 \frac{5 \sqrt{545} \sqrt{79}}{79^{\frac{2}{2}}}$

Step 1.7.6.4

Cancel the common factor of

$2$ .

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Step 1.7.6.4.1

Cancel the common factor.

$5285 - 5245 + 1.96 \frac{5 \sqrt{545} \sqrt{79}}{79^{\frac{2}{2}}}$

Step 1.7.6.4.2

Rewrite the expression.

$5285 - 5245 + 1.96 \frac{5 \sqrt{545} \sqrt{79}}{79^{1}}$

Step 1.7.6.5

Evaluate the exponent.

$5285 - 5245 + 1.96 \frac{5 \sqrt{545} \sqrt{79}}{79}$

Step 1.8

Simplify the numerator.

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Step 1.8.1

Combine using the product rule for radicals.

$5285 - 5245 + 1.96 \frac{5 \sqrt{79 \cdot 545}}{79}$

Step 1.8.2

Multiply

$79$ by

$545$ .

$5285 - 5245 + 1.96 \frac{5 \sqrt{43055}}{79}$

Step 1.9

Multiply

$1.96 \frac{5 \sqrt{43055}}{79}$ .

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Step 1.9.1

Combine

$1.96$ and

$\frac{5 \sqrt{43055}}{79}$ .

$5285 - 5245 + \frac{1.96 (5 \sqrt{43055})}{79}$

Step 1.9.2

Multiply

$5$ by

$1.96$ .

$5285 - 5245 + \frac{9.8 \sqrt{43055}}{79}$

Step 1.9.3

Multiply

$9.8$ by

$\sqrt{43055}$ .

$5285 - 5245 + \frac{2033.47048171}{79}$

Step 1.10

Divide

$2033.47048171$ by

$79$ .

$5285 - 5245 + 25.74013267$

Step 2

Simplify by adding and subtracting.

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Step 2.1

Subtract

$5245$ from

$5285$ .

$40 + 25.74013267$

Step 2.2

Add

$40$ and

$25.74013267$ .

$65.74013267$