Basic Math Examples

$Q = v ({(199 \sin (15))}^{2} - 2 (9.8 (1290)))$

Step 1

Remove parentheses.

$Q = v ({(199 \sin (15))}^{2} - 2 \cdot 9.8 \cdot 1290)$

Step 2

Simplify $v ({(199 \sin (15))}^{2} - 2 \cdot 9.8 \cdot 1290)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

The exact value of $\sin (15)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 2.1.1.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$Q = v ({(199 \sin (45 - 30))}^{2} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.1.2

Separate negation.

$Q = v ({(199 \sin (45 - (30)))}^{2} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.1.3

Apply the difference of angles identity.

$Q = v ({(199 (\sin (45) \cos (30) - \cos (45) \sin (30)))}^{2} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.1.4

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$Q = v ({(199 (\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30)))}^{2} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.1.5

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$Q = v ({(199 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30)))}^{2} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.1.6

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$Q = v ({(199 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30)))}^{2} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.1.7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$Q = v ({(199 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))}^{2} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.1.8

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 2.1.1.8.1

Simplify each term.

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Step 2.1.1.8.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

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Step 2.1.1.8.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$Q = v ({(199 (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))}^{2} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.1.8.1.1.2

Combine using the product rule for radicals.

$Q = v ({(199 (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))}^{2} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.1.8.1.1.3

Multiply $2$ by $3$ .

$Q = v ({(199 (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))}^{2} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.1.8.1.1.4

Multiply $2$ by $2$ .

$Q = v ({(199 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))}^{2} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.1.8.1.2

Multiply $- \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 2.1.1.8.1.2.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{2}}{2}$ .

$Q = v ({(199 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}))}^{2} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.1.8.1.2.2

Multiply $2$ by $2$ .

$Q = v ({(199 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}))}^{2} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.1.8.2

Combine the numerators over the common denominator.

$Q = v ({(199 \frac{\sqrt{6} - \sqrt{2}}{4})}^{2} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.2

Combine $199$ and $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

$Q = v ({(\frac{199 (\sqrt{6} - \sqrt{2})}{4})}^{2} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.3.1

Apply the product rule to $\frac{199 (\sqrt{6} - \sqrt{2})}{4}$ .

$Q = v (\frac{{(199 (\sqrt{6} - \sqrt{2}))}^{2}}{4^{2}} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.3.2

Apply the product rule to $199 (\sqrt{6} - \sqrt{2})$ .

$Q = v (\frac{199^{2} {(\sqrt{6} - \sqrt{2})}^{2}}{4^{2}} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.4

Raise $199$ to the power of $2$ .

$Q = v (\frac{39601 {(\sqrt{6} - \sqrt{2})}^{2}}{4^{2}} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.5

Raise $4$ to the power of $2$ .

$Q = v (\frac{39601 {(\sqrt{6} - \sqrt{2})}^{2}}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.6

Rewrite ${(\sqrt{6} - \sqrt{2})}^{2}$ as $(\sqrt{6} - \sqrt{2}) (\sqrt{6} - \sqrt{2})$ .

$Q = v (\frac{39601 ((\sqrt{6} - \sqrt{2}) (\sqrt{6} - \sqrt{2}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.7

Expand $(\sqrt{6} - \sqrt{2}) (\sqrt{6} - \sqrt{2})$ using the FOIL Method.

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Step 2.1.7.1

Apply the distributive property.

$Q = v (\frac{39601 (\sqrt{6} (\sqrt{6} - \sqrt{2}) - \sqrt{2} (\sqrt{6} - \sqrt{2}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.7.2

Apply the distributive property.

$Q = v (\frac{39601 (\sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} (\sqrt{6} - \sqrt{2}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.7.3

Apply the distributive property.

$Q = v (\frac{39601 (\sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8

Simplify and combine like terms.

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Step 2.1.8.1

Simplify each term.

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Step 2.1.8.1.1

Combine using the product rule for radicals.

$Q = v (\frac{39601 (\sqrt{6 \cdot 6} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.2

Multiply $6$ by $6$ .

$Q = v (\frac{39601 (\sqrt{36} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.3

Rewrite $36$ as $6^{2}$ .

$Q = v (\frac{39601 (\sqrt{6^{2}} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.4

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

$Q = v (\frac{39601 (6 + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.5

Multiply $\sqrt{6} (- \sqrt{2})$ .

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Step 2.1.8.1.5.1

Combine using the product rule for radicals.

$Q = v (\frac{39601 (6 - \sqrt{6 \cdot 2} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.5.2

Multiply $6$ by $2$ .

$Q = v (\frac{39601 (6 - \sqrt{12} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.6

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.1.8.1.6.1

Factor $4$ out of $12$ .

$Q = v (\frac{39601 (6 - \sqrt{4 (3)} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.6.2

Rewrite $4$ as $2^{2}$ .

$Q = v (\frac{39601 (6 - \sqrt{2^{2} \cdot 3} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.7

Pull terms out from under the radical.

$Q = v (\frac{39601 (6 - (2 \sqrt{3}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.8

Multiply $2$ by $- 1$ .

$Q = v (\frac{39601 (6 - 2 \sqrt{3} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.9

Multiply $- \sqrt{2} \sqrt{6}$ .

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Step 2.1.8.1.9.1

Combine using the product rule for radicals.

$Q = v (\frac{39601 (6 - 2 \sqrt{3} - \sqrt{6 \cdot 2} - \sqrt{2} (- \sqrt{2}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.9.2

Multiply $6$ by $2$ .

$Q = v (\frac{39601 (6 - 2 \sqrt{3} - \sqrt{12} - \sqrt{2} (- \sqrt{2}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.10

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.1.8.1.10.1

Factor $4$ out of $12$ .

$Q = v (\frac{39601 (6 - 2 \sqrt{3} - \sqrt{4 (3)} - \sqrt{2} (- \sqrt{2}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.10.2

Rewrite $4$ as $2^{2}$ .

$Q = v (\frac{39601 (6 - 2 \sqrt{3} - \sqrt{2^{2} \cdot 3} - \sqrt{2} (- \sqrt{2}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.11

Pull terms out from under the radical.

$Q = v (\frac{39601 (6 - 2 \sqrt{3} - (2 \sqrt{3}) - \sqrt{2} (- \sqrt{2}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.12

Multiply $2$ by $- 1$ .

$Q = v (\frac{39601 (6 - 2 \sqrt{3} - 2 \sqrt{3} - \sqrt{2} (- \sqrt{2}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.13

Multiply $- \sqrt{2} (- \sqrt{2})$ .

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Step 2.1.8.1.13.1

Multiply $- 1$ by $- 1$ .

$Q = v (\frac{39601 (6 - 2 \sqrt{3} - 2 \sqrt{3} + 1 \sqrt{2} \sqrt{2})}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.13.2

Multiply $\sqrt{2}$ by $1$ .

$Q = v (\frac{39601 (6 - 2 \sqrt{3} - 2 \sqrt{3} + \sqrt{2} \sqrt{2})}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.13.3

Raise $\sqrt{2}$ to the power of $1$ .

$Q = v (\frac{39601 (6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{1} \sqrt{2})}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.13.4

Raise $\sqrt{2}$ to the power of $1$ .

$Q = v (\frac{39601 (6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{1} {\sqrt{2}}^{1})}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.13.5

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

$Q = v (\frac{39601 (6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{1 + 1})}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.13.6

Add $1$ and $1$ .

$Q = v (\frac{39601 (6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{2})}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.14

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

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Step 2.1.8.1.14.1

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

$Q = v (\frac{39601 (6 - 2 \sqrt{3} - 2 \sqrt{3} + {(2^{\frac{1}{2}})}^{2})}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.14.2

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

$Q = v (\frac{39601 (6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{\frac{1}{2} \cdot 2})}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.14.3

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

$Q = v (\frac{39601 (6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{\frac{2}{2}})}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.14.4

Cancel the common factor of $2$ .

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Step 2.1.8.1.14.4.1

Cancel the common factor.

$Q = v (\frac{39601 (6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{\frac{2}{2}})}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.14.4.2

Rewrite the expression.

$Q = v (\frac{39601 (6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{1})}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.1.14.5

Evaluate the exponent.

$Q = v (\frac{39601 (6 - 2 \sqrt{3} - 2 \sqrt{3} + 2)}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.2

Add $6$ and $2$ .

$Q = v (\frac{39601 (8 - 2 \sqrt{3} - 2 \sqrt{3})}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.8.3

Subtract $2 \sqrt{3}$ from $- 2 \sqrt{3}$ .

$Q = v (\frac{39601 (8 - 4 \sqrt{3})}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.9

Cancel the common factor of $8 - 4 \sqrt{3}$ and $16$ .

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Step 2.1.9.1

Factor $4$ out of $39601 (8 - 4 \sqrt{3})$ .

$Q = v (\frac{4 (39601 (2 - \sqrt{3}))}{16} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.9.2

Cancel the common factors.

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Step 2.1.9.2.1

Factor $4$ out of $16$ .

$Q = v (\frac{4 (39601 (2 - \sqrt{3}))}{4 (4)} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.9.2.2

Cancel the common factor.

$Q = v (\frac{4 (39601 (2 - \sqrt{3}))}{4 \cdot 4} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.9.2.3

Rewrite the expression.

$Q = v (\frac{39601 (2 - \sqrt{3})}{4} - 2 \cdot 9.8 \cdot 1290)$

Step 2.1.10

Multiply $- 2 \cdot 9.8 \cdot 1290$ .

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Step 2.1.10.1

Multiply $- 2$ by $9.8$ .

$Q = v (\frac{39601 (2 - \sqrt{3})}{4} - 19.6 \cdot 1290)$

Step 2.1.10.2

Multiply $- 19.6$ by $1290$ .

$Q = v (\frac{39601 (2 - \sqrt{3})}{4} - 25284)$

Step 2.2

To write $- 25284$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$Q = v (\frac{39601 (2 - \sqrt{3})}{4} - 25284 \cdot \frac{4}{4})$

Step 2.3

Combine fractions.

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Step 2.3.1

Combine $- 25284$ and $\frac{4}{4}$ .

$Q = v (\frac{39601 (2 - \sqrt{3})}{4} + \frac{- 25284 \cdot 4}{4})$

Step 2.3.2

Combine the numerators over the common denominator.

$Q = v \frac{39601 (2 - \sqrt{3}) - 25284 \cdot 4}{4}$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Apply the distributive property.

$Q = v \frac{39601 \cdot 2 + 39601 (- \sqrt{3}) - 25284 \cdot 4}{4}$

Step 2.4.2

Multiply $39601$ by $2$ .

$Q = v \frac{79202 + 39601 (- \sqrt{3}) - 25284 \cdot 4}{4}$

Step 2.4.3

Multiply $- 1$ by $39601$ .

$Q = v \frac{79202 - 39601 \sqrt{3} - 25284 \cdot 4}{4}$

Step 2.4.4

Multiply $- 25284$ by $4$ .

$Q = v \frac{79202 - 39601 \sqrt{3} - 101136}{4}$

Step 2.4.5

Subtract $101136$ from $79202$ .

$Q = v \frac{- 21934 - 39601 \sqrt{3}}{4}$

Step 2.4.6

Subtract $39601 \sqrt{3}$ from $- 21934$ .

$Q = v \frac{- 90524.94403053}{4}$

Step 2.5

Simplify the expression.

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Step 2.5.1

Divide $- 90524.94403053$ by $4$ .

$Q = v \cdot - 22631.23600763$

Step 2.5.2

Move $- 22631.23600763$ to the left of $v$ .

$Q = - 22631.23600763 v$