Basic Math Examples

$1.1 = v \sin (45) (\frac{4.4}{v \cos (45)}) - 4.9 \frac{4.4}{v \cos (45)}$

Step 1

Rewrite the equation as $v \sin (45) (\frac{4.4}{v \cos (45)}) - 4.9 \frac{4.4}{v \cos (45)} = 1.1$ .

$v \sin (45) (\frac{4.4}{v \cos (45)}) - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2

Simplify the left side.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$v \sin (45) \frac{4.4}{v \frac{\sqrt{2}}{2}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.2

Combine $v$ and $\frac{\sqrt{2}}{2}$ .

$v \sin (45) \frac{4.4}{\frac{v \sqrt{2}}{2}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.3

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

$v \frac{\sqrt{2}}{2} \frac{4.4}{\frac{v \sqrt{2}}{2}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.4

Combine $v$ and $\frac{\sqrt{2}}{2}$ .

$\frac{v \sqrt{2}}{2} \cdot \frac{4.4}{\frac{v \sqrt{2}}{2}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{v \sqrt{2}}{2} (4.4 \frac{2}{v \sqrt{2}}) - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.6

Rewrite using the commutative property of multiplication.

$4.4 \frac{v \sqrt{2}}{2} \frac{2}{v \sqrt{2}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.7

Multiply $4.4 \frac{v \sqrt{2}}{2}$ .

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Step 2.1.7.1

Combine $4.4$ and $\frac{v \sqrt{2}}{2}$ .

$\frac{4.4 (v \sqrt{2})}{2} \cdot \frac{2}{v \sqrt{2}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.7.2

Multiply $\sqrt{2}$ by $4.4$ .

$\frac{6.22253967 v}{2} \cdot \frac{2}{v \sqrt{2}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.8

Cancel the common factor of $v$ .

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Step 2.1.8.1

Factor $v$ out of $6.22253967 v$ .

$\frac{v \cdot 6.22253967}{2} \cdot \frac{2}{v \sqrt{2}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.8.2

Factor $v$ out of $v \sqrt{2}$ .

$\frac{v \cdot 6.22253967}{2} \cdot \frac{2}{v (\sqrt{2})} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.8.3

Cancel the common factor.

$\frac{v \cdot 6.22253967}{2} \cdot \frac{2}{v \sqrt{2}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.8.4

Rewrite the expression.

$\frac{6.22253967}{2} \cdot \frac{2}{\sqrt{2}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.9

Cancel the common factor of $2$ .

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Step 2.1.9.1

Cancel the common factor.

$\frac{6.22253967}{2} \cdot \frac{2}{\sqrt{2}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.9.2

Rewrite the expression.

$6.22253967 \frac{1}{\sqrt{2}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.10

Combine $6.22253967$ and $\frac{1}{\sqrt{2}}$ .

$\frac{6.22253967}{\sqrt{2}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.11

Multiply $\frac{6.22253967}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{6.22253967}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.12

Combine and simplify the denominator.

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Step 2.1.12.1

Multiply $\frac{6.22253967}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{6.22253967 \sqrt{2}}{\sqrt{2} \sqrt{2}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.12.2

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

$\frac{6.22253967 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.12.3

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

$\frac{6.22253967 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.12.4

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

$\frac{6.22253967 \sqrt{2}}{{\sqrt{2}}^{1 + 1}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.12.5

Add $1$ and $1$ .

$\frac{6.22253967 \sqrt{2}}{{\sqrt{2}}^{2}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.12.6

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

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Step 2.1.12.6.1

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

$\frac{6.22253967 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.12.6.2

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

$\frac{6.22253967 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.12.6.3

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

$\frac{6.22253967 \sqrt{2}}{2^{\frac{2}{2}}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.12.6.4

Cancel the common factor of $2$ .

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Step 2.1.12.6.4.1

Cancel the common factor.

$\frac{6.22253967 \sqrt{2}}{2^{\frac{2}{2}}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.12.6.4.2

Rewrite the expression.

$\frac{6.22253967 \sqrt{2}}{2^{1}} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.12.6.5

Evaluate the exponent.

$\frac{6.22253967 \sqrt{2}}{2} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.13

Multiply $6.22253967$ by $\sqrt{2}$ .

$\frac{8.8}{2} - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.14

Divide $8.8$ by $2$ .

$4.4 - 4.9 \frac{4.4}{v \cos (45)} = 1.1$

Step 2.1.15

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

$4.4 - 4.9 \frac{4.4}{v \frac{\sqrt{2}}{2}} = 1.1$

Step 2.1.16

Combine $v$ and $\frac{\sqrt{2}}{2}$ .

$4.4 - 4.9 \frac{4.4}{\frac{v \sqrt{2}}{2}} = 1.1$

Step 2.1.17

Multiply the numerator by the reciprocal of the denominator.

$4.4 - 4.9 (4.4 \frac{2}{v \sqrt{2}}) = 1.1$

Step 2.1.18

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

$4.4 - 4.9 (4.4 (\frac{2}{v \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})) = 1.1$

Step 2.1.19

Combine and simplify the denominator.

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Step 2.1.19.1

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

$4.4 - 4.9 (4.4 \frac{2 \sqrt{2}}{v \sqrt{2} \sqrt{2}}) = 1.1$

Step 2.1.19.2

Move $\sqrt{2}$ .

$4.4 - 4.9 (4.4 \frac{2 \sqrt{2}}{v (\sqrt{2} \sqrt{2})}) = 1.1$

Step 2.1.19.3

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

$4.4 - 4.9 (4.4 \frac{2 \sqrt{2}}{v ({\sqrt{2}}^{1} \sqrt{2})}) = 1.1$

Step 2.1.19.4

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

$4.4 - 4.9 (4.4 \frac{2 \sqrt{2}}{v ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}) = 1.1$

Step 2.1.19.5

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

$4.4 - 4.9 (4.4 \frac{2 \sqrt{2}}{v {\sqrt{2}}^{1 + 1}}) = 1.1$

Step 2.1.19.6

Add $1$ and $1$ .

$4.4 - 4.9 (4.4 \frac{2 \sqrt{2}}{v {\sqrt{2}}^{2}}) = 1.1$

Step 2.1.19.7

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

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Step 2.1.19.7.1

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

$4.4 - 4.9 (4.4 \frac{2 \sqrt{2}}{v {(2^{\frac{1}{2}})}^{2}}) = 1.1$

Step 2.1.19.7.2

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

$4.4 - 4.9 (4.4 \frac{2 \sqrt{2}}{v \cdot 2^{\frac{1}{2} \cdot 2}}) = 1.1$

Step 2.1.19.7.3

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

$4.4 - 4.9 (4.4 \frac{2 \sqrt{2}}{v \cdot 2^{\frac{2}{2}}}) = 1.1$

Step 2.1.19.7.4

Cancel the common factor of $2$ .

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Step 2.1.19.7.4.1

Cancel the common factor.

$4.4 - 4.9 (4.4 \frac{2 \sqrt{2}}{v \cdot 2^{\frac{2}{2}}}) = 1.1$

Step 2.1.19.7.4.2

Rewrite the expression.

$4.4 - 4.9 (4.4 \frac{2 \sqrt{2}}{v \cdot 2^{1}}) = 1.1$

Step 2.1.19.7.5

Evaluate the exponent.

$4.4 - 4.9 (4.4 \frac{2 \sqrt{2}}{v \cdot 2}) = 1.1$

Step 2.1.20

Cancel the common factor of $2$ .

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Step 2.1.20.1

Cancel the common factor.

$4.4 - 4.9 (4.4 \frac{2 \sqrt{2}}{v \cdot 2}) = 1.1$

Step 2.1.20.2

Rewrite the expression.

$4.4 - 4.9 (4.4 \frac{\sqrt{2}}{v}) = 1.1$

Step 2.1.21

Multiply $4.4 \frac{\sqrt{2}}{v}$ .

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Step 2.1.21.1

Combine $4.4$ and $\frac{\sqrt{2}}{v}$ .

$4.4 - 4.9 \frac{4.4 \sqrt{2}}{v} = 1.1$

Step 2.1.21.2

Multiply $4.4$ by $\sqrt{2}$ .

$4.4 - 4.9 \frac{6.22253967}{v} = 1.1$

Step 2.1.22

Multiply $- 4.9 \frac{6.22253967}{v}$ .

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Step 2.1.22.1

Combine $- 4.9$ and $\frac{6.22253967}{v}$ .

$4.4 + \frac{- 4.9 \cdot 6.22253967}{v} = 1.1$

Step 2.1.22.2

Multiply $- 4.9$ by $6.22253967$ .

$4.4 + \frac{- 30.4904444}{v} = 1.1$

Step 2.1.23

Move the negative in front of the fraction.

$4.4 - \frac{30.4904444}{v} = 1.1$

Step 3

Move all terms not containing $v$ to the right side of the equation.

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Step 3.1

Subtract $4.4$ from both sides of the equation.

$- \frac{30.4904444}{v} = 1.1 - 4.4$

Step 3.2

Subtract $4.4$ from $1.1$ .

$- \frac{30.4904444}{v} = - 3.3$

Step 4

Find the LCD of the terms in the equation.

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Step 4.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$v, 1$

Step 4.2

The LCM of one and any expression is the expression.

$v$

Step 5

Multiply each term in $- \frac{30.4904444}{v} = - 3.3$ by $v$ to eliminate the fractions.

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Step 5.1

Multiply each term in $- \frac{30.4904444}{v} = - 3.3$ by $v$ .

$- \frac{30.4904444}{v} v = - 3.3 v$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $v$ .

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Step 5.2.1.1

Move the leading negative in $- \frac{30.4904444}{v}$ into the numerator.

$\frac{- 30.4904444}{v} v = - 3.3 v$

Step 5.2.1.2

Cancel the common factor.

$\frac{- 30.4904444}{v} v = - 3.3 v$

Step 5.2.1.3

Rewrite the expression.

$- 30.4904444 = - 3.3 v$

Step 6

Solve the equation.

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Step 6.1

Rewrite the equation as $- 3.3 v = - 30.4904444$ .

$- 3.3 v = - 30.4904444$

Step 6.2

Divide each term in $- 3.3 v = - 30.4904444$ by $- 3.3$ and simplify.

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Step 6.2.1

Divide each term in $- 3.3 v = - 30.4904444$ by $- 3.3$ .

$\frac{- 3.3 v}{- 3.3} = \frac{- 30.4904444}{- 3.3}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $- 3.3$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{- 3.3 v}{- 3.3} = \frac{- 30.4904444}{- 3.3}$

Step 6.2.2.1.2

Divide $v$ by $1$ .

$v = \frac{- 30.4904444}{- 3.3}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $- 30.4904444$ by $- 3.3$ .

$v = 9.2395286$