Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Combine the numerators over the common denominator.

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

Step 1.2

Simplify the numerator.

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Step 1.2.1

Raise $155$ to the power of $2$ .

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

Step 1.2.2

Raise $210$ to the power of $2$ .

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

Step 1.2.3

Add $24025$ and $44100$ .

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

Step 1.3

Cancel the common factor of $68125$ and $395$ .

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Step 1.3.1

Factor $5$ out of $68125$ .

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

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $5$ out of $395$ .

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

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 $25.74013267$ .

$65.74013267$