Basic Math Examples

$154 = \frac{(22 \div 7) (r^{2})}{\frac{\frac{2}{3}}{4}}$

Step 1

Rewrite the equation as $\frac{22 \div 7 r^{2}}{\frac{\frac{2}{3}}{4}} = 154$ .

$\frac{22 \div 7 r^{2}}{\frac{\frac{2}{3}}{4}} = 154$

Step 2

Multiply both sides by $\frac{\frac{2}{3}}{4}$ .

$\frac{22 \div 7 r^{2}}{\frac{\frac{2}{3}}{4}} \cdot \frac{\frac{2}{3}}{4} = 154 \frac{\frac{2}{3}}{4}$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{22 \div 7 r^{2}}{\frac{\frac{2}{3}}{4}} \cdot \frac{\frac{2}{3}}{4}$ .

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Step 3.1.1.1

Combine fractions.

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Step 3.1.1.1.1

Combine.

$\frac{22 \div 7 r^{2} \cdot \frac{2}{3}}{\frac{\frac{2}{3}}{4} \cdot 4} = 154 \frac{\frac{2}{3}}{4}$

Step 3.1.1.1.2

Multiply $22 \div 7 r^{2}$ by $\frac{2}{3}$ .

$\frac{\frac{22 \cdot 2}{7 r^{2} \cdot 3}}{\frac{\frac{2}{3}}{4} \cdot 4} = 154 \frac{\frac{2}{3}}{4}$

Step 3.1.1.1.3

Combine $\frac{\frac{2}{3}}{4}$ and $4$ .

$\frac{\frac{22 \cdot 2}{7 r^{2} \cdot 3}}{\frac{\frac{2}{3} \cdot 4}{4}} = 154 \frac{\frac{2}{3}}{4}$

Step 3.1.1.1.4

Multiply.

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Step 3.1.1.1.4.1

Multiply $22$ by $2$ .

$\frac{\frac{44}{7 r^{2} \cdot 3}}{\frac{\frac{2}{3} \cdot 4}{4}} = 154 \frac{\frac{2}{3}}{4}$

Step 3.1.1.1.4.2

Multiply $3$ by $7$ .

$\frac{\frac{44}{21 r^{2}}}{\frac{\frac{2}{3} \cdot 4}{4}} = 154 \frac{\frac{2}{3}}{4}$

Step 3.1.1.1.5

Combine $\frac{2}{3}$ and $4$ .

$\frac{\frac{44}{21 r^{2}}}{\frac{\frac{2 \cdot 4}{3}}{4}} = 154 \frac{\frac{2}{3}}{4}$

Step 3.1.1.1.6

Multiply $2$ by $4$ .

$\frac{\frac{44}{21 r^{2}}}{\frac{\frac{8}{3}}{4}} = 154 \frac{\frac{2}{3}}{4}$

Step 3.1.1.2

Simplify the denominator.

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Step 3.1.1.2.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{\frac{44}{21 r^{2}}}{\frac{8}{3} \cdot \frac{1}{4}} = 154 \frac{\frac{2}{3}}{4}$

Step 3.1.1.2.2

Cancel the common factor of $4$ .

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Step 3.1.1.2.2.1

Factor $4$ out of $8$ .

$\frac{\frac{44}{21 r^{2}}}{\frac{4 (2)}{3} \cdot \frac{1}{4}} = 154 \frac{\frac{2}{3}}{4}$

Step 3.1.1.2.2.2

Cancel the common factor.

$\frac{\frac{44}{21 r^{2}}}{\frac{4 \cdot 2}{3} \cdot \frac{1}{4}} = 154 \frac{\frac{2}{3}}{4}$

Step 3.1.1.2.2.3

Rewrite the expression.

$\frac{\frac{44}{21 r^{2}}}{\frac{2}{3}} = 154 \frac{\frac{2}{3}}{4}$

Step 3.1.1.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{44}{21 r^{2}} \cdot \frac{3}{2} = 154 \frac{\frac{2}{3}}{4}$

Step 3.1.1.4

Combine.

$\frac{44 \cdot 3}{21 r^{2} \cdot 2} = 154 \frac{\frac{2}{3}}{4}$

Step 3.1.1.5

Cancel the common factor of $44$ and $2$ .

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Step 3.1.1.5.1

Factor $2$ out of $44 \cdot 3$ .

$\frac{2 (22 \cdot 3)}{21 r^{2} \cdot 2} = 154 \frac{\frac{2}{3}}{4}$

Step 3.1.1.5.2

Cancel the common factors.

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Step 3.1.1.5.2.1

Factor $2$ out of $21 r^{2} \cdot 2$ .

$\frac{2 (22 \cdot 3)}{2 \cdot (21 r^{2})} = 154 \frac{\frac{2}{3}}{4}$

Step 3.1.1.5.2.2

Cancel the common factor.

$\frac{2 (22 \cdot 3)}{2 \cdot (21 r^{2})} = 154 \frac{\frac{2}{3}}{4}$

Step 3.1.1.5.2.3

Rewrite the expression.

$\frac{22 \cdot 3}{21 r^{2}} = 154 \frac{\frac{2}{3}}{4}$

Step 3.1.1.6

Cancel the common factor of $3$ and $21$ .

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Step 3.1.1.6.1

Factor $3$ out of $22 \cdot 3$ .

$\frac{3 \cdot 22}{21 r^{2}} = 154 \frac{\frac{2}{3}}{4}$

Step 3.1.1.6.2

Cancel the common factors.

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Step 3.1.1.6.2.1

Factor $3$ out of $21 r^{2}$ .

$\frac{3 \cdot 22}{3 (7 r^{2})} = 154 \frac{\frac{2}{3}}{4}$

Step 3.1.1.6.2.2

Cancel the common factor.

$\frac{3 \cdot 22}{3 (7 r^{2})} = 154 \frac{\frac{2}{3}}{4}$

Step 3.1.1.6.2.3

Rewrite the expression.

$\frac{22}{7 r^{2}} = 154 \frac{\frac{2}{3}}{4}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $154 \frac{\frac{2}{3}}{4}$ .

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Step 3.2.1.1

Cancel the common factor of $2$ .

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Step 3.2.1.1.1

Factor $2$ out of $154$ .

$\frac{22}{7 r^{2}} = 2 (77) \frac{\frac{2}{3}}{4}$

Step 3.2.1.1.2

Factor $2$ out of $4$ .

$\frac{22}{7 r^{2}} = 2 \cdot 77 \frac{\frac{2}{3}}{2 \cdot 2}$

Step 3.2.1.1.3

Cancel the common factor.

$\frac{22}{7 r^{2}} = 2 \cdot 77 \frac{\frac{2}{3}}{2 \cdot 2}$

Step 3.2.1.1.4

Rewrite the expression.

$\frac{22}{7 r^{2}} = 77 \frac{\frac{2}{3}}{2}$

Step 3.2.1.2

Combine $77$ and $\frac{\frac{2}{3}}{2}$ .

$\frac{22}{7 r^{2}} = \frac{77 (\frac{2}{3})}{2}$

Step 3.2.1.3

Combine $77$ and $\frac{2}{3}$ .

$\frac{22}{7 r^{2}} = \frac{\frac{77 \cdot 2}{3}}{2}$

Step 3.2.1.4

Multiply $77$ by $2$ .

$\frac{22}{7 r^{2}} = \frac{\frac{154}{3}}{2}$

Step 3.2.1.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{22}{7 r^{2}} = \frac{154}{3} \cdot \frac{1}{2}$

Step 3.2.1.6

Cancel the common factor of $2$ .

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Step 3.2.1.6.1

Factor $2$ out of $154$ .

$\frac{22}{7 r^{2}} = \frac{2 (77)}{3} \cdot \frac{1}{2}$

Step 3.2.1.6.2

Cancel the common factor.

$\frac{22}{7 r^{2}} = \frac{2 \cdot 77}{3} \cdot \frac{1}{2}$

Step 3.2.1.6.3

Rewrite the expression.

$\frac{22}{7 r^{2}} = \frac{77}{3}$

Step 4

Solve for $r$ .

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Step 4.1

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$22 \cdot 3 = 7 r^{2} \cdot 77$

Step 4.2

Solve the equation for $r$ .

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Step 4.2.1

Rewrite the equation as $7 r^{2} \cdot 77 = 22 \cdot 3$ .

$7 r^{2} \cdot 77 = 22 \cdot 3$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Multiply $77$ by $7$ .

$539 r^{2} = 22 \cdot 3$

Step 4.2.2.2

Multiply $22$ by $3$ .

$539 r^{2} = 66$

Step 4.2.3

Divide each term in $539 r^{2} = 66$ by $539$ and simplify.

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Step 4.2.3.1

Divide each term in $539 r^{2} = 66$ by $539$ .

$\frac{539 r^{2}}{539} = \frac{66}{539}$

Step 4.2.3.2

Simplify the left side.

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Step 4.2.3.2.1

Cancel the common factor of $539$ .

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Step 4.2.3.2.1.1

Cancel the common factor.

$\frac{539 r^{2}}{539} = \frac{66}{539}$

Step 4.2.3.2.1.2

Divide $r^{2}$ by $1$ .

$r^{2} = \frac{66}{539}$

Step 4.2.3.3

Simplify the right side.

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Step 4.2.3.3.1

Cancel the common factor of $66$ and $539$ .

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Step 4.2.3.3.1.1

Factor $11$ out of $66$ .

$r^{2} = \frac{11 (6)}{539}$

Step 4.2.3.3.1.2

Cancel the common factors.

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Step 4.2.3.3.1.2.1

Factor $11$ out of $539$ .

$r^{2} = \frac{11 \cdot 6}{11 \cdot 49}$

Step 4.2.3.3.1.2.2

Cancel the common factor.

$r^{2} = \frac{11 \cdot 6}{11 \cdot 49}$

Step 4.2.3.3.1.2.3

Rewrite the expression.

$r^{2} = \frac{6}{49}$

Step 4.2.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$r = \pm \sqrt{\frac{6}{49}}$

Step 4.2.5

Simplify $\pm \sqrt{\frac{6}{49}}$ .

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Step 4.2.5.1

Rewrite $\sqrt{\frac{6}{49}}$ as $\frac{\sqrt{6}}{\sqrt{49}}$ .

$r = \pm \frac{\sqrt{6}}{\sqrt{49}}$

Step 4.2.5.2

Simplify the denominator.

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Step 4.2.5.2.1

Rewrite $49$ as $7^{2}$ .

$r = \pm \frac{\sqrt{6}}{\sqrt{7^{2}}}$

Step 4.2.5.2.2

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

$r = \pm \frac{\sqrt{6}}{7}$

Step 4.2.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.2.6.1

First, use the positive value of the $\pm$ to find the first solution.

$r = \frac{\sqrt{6}}{7}$

Step 4.2.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$r = - \frac{\sqrt{6}}{7}$

Step 4.2.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$r = \frac{\sqrt{6}}{7}, - \frac{\sqrt{6}}{7}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$r = \frac{\sqrt{6}}{7}, - \frac{\sqrt{6}}{7}$

Decimal Form:

$r = 0.34992710 \dots, - 0.34992710 \dots$