Trigonometry Examples

$4.634 = \frac{k^{3.5}}{\sqrt{110.25}}$

Step 1

Rewrite the equation as $\frac{k^{3.5}}{\sqrt{110.25}} = 4.634$ .

$\frac{k^{3.5}}{\sqrt{110.25}} = 4.634$

Step 2

Multiply both sides of the equation by $\sqrt{110.25}$ .

$\sqrt{110.25} \frac{k^{3.5}}{\sqrt{110.25}} = \sqrt{110.25} \cdot 4.634$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $\sqrt{110.25}$ .

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Step 3.1.1.1

Cancel the common factor.

$\sqrt{110.25} \frac{k^{3.5}}{\sqrt{110.25}} = \sqrt{110.25} \cdot 4.634$

Step 3.1.1.2

Rewrite the expression.

$k^{3.5} = \sqrt{110.25} \cdot 4.634$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\sqrt{110.25} \cdot 4.634$ .

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Step 3.2.1.1

Rewrite $110.25$ as ${10.5}^{2}$ .

$k^{3.5} = \sqrt{{10.5}^{2}} \cdot 4.634$

Step 3.2.1.2

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

$k^{3.5} = 10.5 \cdot 4.634$

Step 3.2.1.3

Multiply $10.5$ by $4.634$ .

$k^{3.5} = 48.657$

Step 4

Convert the decimal exponent to a fractional exponent.

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Step 4.1

Convert the decimal number to a fraction by placing the decimal number over a power of ten. Since there is $1$ number to the right of the decimal point, place the decimal number over $10^{1}$ $(10)$ . Next, add the whole number to the left of the decimal.

$k^{3 \frac{5}{10}} = 48.657$

Step 4.2

Reduce the fractional part of the mixed number.

$k^{3 \frac{1}{2}} = 48.657$

Step 4.3

Convert $3 \frac{1}{2}$ to an improper fraction.

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Step 4.3.1

A mixed number is an addition of its whole and fractional parts.

$k^{3 + \frac{1}{2}} = 48.657$

Step 4.3.2

Add $3$ and $\frac{1}{2}$ .

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Step 4.3.2.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$k^{3 \cdot \frac{2}{2} + \frac{1}{2}} = 48.657$

Step 4.3.2.2

Combine $3$ and $\frac{2}{2}$ .

$k^{\frac{3 \cdot 2}{2} + \frac{1}{2}} = 48.657$

Step 4.3.2.3

Combine the numerators over the common denominator.

$k^{\frac{3 \cdot 2 + 1}{2}} = 48.657$

Step 4.3.2.4

Simplify the numerator.

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Step 4.3.2.4.1

Multiply $3$ by $2$ .

$k^{\frac{6 + 1}{2}} = 48.657$

Step 4.3.2.4.2

Add $6$ and $1$ .

$k^{\frac{7}{2}} = 48.657$

Step 5

Raise each side of the equation to the power of $\frac{1}{3.5}$ to eliminate the fractional exponent on the left side.

${(k^{\frac{7}{2}})}^{\frac{1}{3.5}} = {48.657}^{\frac{1}{3.5}}$

Step 6

Simplify the exponent.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify ${(k^{\frac{7}{2}})}^{\frac{1}{3.5}}$ .

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Step 6.1.1.1

Multiply the exponents in ${(k^{\frac{7}{2}})}^{\frac{1}{3.5}}$ .

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Step 6.1.1.1.1

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

$k^{\frac{7}{2} \cdot \frac{1}{3.5}} = {48.657}^{\frac{1}{3.5}}$

Step 6.1.1.1.2

Cancel the common factor of $3.5$ .

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Step 6.1.1.1.2.1

Factor $3.5$ out of $7$ .

$k^{\frac{3.5 (2)}{2} \cdot \frac{1}{3.5}} = {48.657}^{\frac{1}{3.5}}$

Step 6.1.1.1.2.2

Cancel the common factor.

$k^{\frac{3.5 \cdot 2}{2} \cdot \frac{1}{3.5}} = {48.657}^{\frac{1}{3.5}}$

Step 6.1.1.1.2.3

Rewrite the expression.

$k^{\frac{2}{2}} = {48.657}^{\frac{1}{3.5}}$

Step 6.1.1.1.3

Divide $2$ by $2$ .

$k^{1} = {48.657}^{\frac{1}{3.5}}$

Step 6.1.1.2

Simplify.

$k = {48.657}^{\frac{1}{3.5}}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify ${48.657}^{\frac{1}{3.5}}$ .

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Step 6.2.1.1

Divide $1$ by $3.5$ .

$k = {48.657}^{0.\overline{285714}}$

Step 6.2.1.2

Raise $48.657$ to the power of $0.\overline{285714}$ .

$k = 3.03418126$