Algebra Examples

$6 = 2 (\frac{22}{7}) \sqrt{\frac{L}{32}}$

Step 1

Rewrite the equation as $2 (\frac{22}{7}) \cdot \sqrt{\frac{L}{32}} = 6$ .

$2 (\frac{22}{7}) \cdot \sqrt{\frac{L}{32}} = 6$

Step 2

Multiply $2 (\frac{22}{7})$ .

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Step 2.1

Combine $2$ and $\frac{22}{7}$ .

$\frac{2 \cdot 22}{7} \cdot \sqrt{\frac{L}{32}} = 6$

Step 2.2

Multiply $2$ by $22$ .

$\frac{44}{7} \cdot \sqrt{\frac{L}{32}} = 6$

Step 3

Multiply each term in $\frac{44}{7} \cdot \sqrt{\frac{L}{32}} = 6$ by $\frac{7}{44}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{44}{7} \cdot \sqrt{\frac{L}{32}} = 6$ by $\frac{7}{44}$ .

$\frac{44}{7} \cdot \sqrt{\frac{L}{32}} \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2

Simplify the left side.

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Step 3.2.1

Rewrite $\frac{L}{32}$ as ${(\frac{1}{4})}^{2} \frac{L}{2}$ .

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Step 3.2.1.1

Factor the perfect power $1^{2}$ out of $L$ .

$\frac{44}{7} \cdot \sqrt{\frac{1^{2} L}{32}} \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.1.2

Factor the perfect power $4^{2}$ out of $32$ .

$\frac{44}{7} \cdot \sqrt{\frac{1^{2} L}{4^{2} \cdot 2}} \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.1.3

Rearrange the fraction $\frac{1^{2} L}{4^{2} \cdot 2}$ .

$\frac{44}{7} \cdot \sqrt{{(\frac{1}{4})}^{2} \frac{L}{2}} \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.2

Pull terms out from under the radical.

$\frac{44}{7} \cdot (\frac{1}{4} \sqrt{\frac{L}{2}}) \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.3

Rewrite $\sqrt{\frac{L}{2}}$ as $\frac{\sqrt{L}}{\sqrt{2}}$ .

$\frac{44}{7} \cdot (\frac{1}{4} \cdot \frac{\sqrt{L}}{\sqrt{2}}) \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.4

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

$\frac{44}{7} \cdot (\frac{1}{4} (\frac{\sqrt{L}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})) \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.5

Combine and simplify the denominator.

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Step 3.2.5.1

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

$\frac{44}{7} \cdot (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{\sqrt{2} \sqrt{2}}) \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.5.2

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

$\frac{44}{7} \cdot (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}) \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.5.3

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

$\frac{44}{7} \cdot (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}) \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.5.4

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

$\frac{44}{7} \cdot (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}) \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.5.5

Add $1$ and $1$ .

$\frac{44}{7} \cdot (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{{\sqrt{2}}^{2}}) \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.5.6

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

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Step 3.2.5.6.1

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

$\frac{44}{7} \cdot (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}) \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.5.6.2

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

$\frac{44}{7} \cdot (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}) \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.5.6.3

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

$\frac{44}{7} \cdot (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{2^{\frac{2}{2}}}) \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.5.6.4

Cancel the common factor of $2$ .

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Step 3.2.5.6.4.1

Cancel the common factor.

$\frac{44}{7} \cdot (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{2^{\frac{2}{2}}}) \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.5.6.4.2

Rewrite the expression.

$\frac{44}{7} \cdot (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{2^{1}}) \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.5.6.5

Evaluate the exponent.

$\frac{44}{7} \cdot (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{2}) \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.6

Combine using the product rule for radicals.

$\frac{44}{7} \cdot (\frac{1}{4} \cdot \frac{\sqrt{L \cdot 2}}{2}) \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.7

Multiply $\frac{1}{4} \cdot \frac{\sqrt{L \cdot 2}}{2}$ .

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Step 3.2.7.1

Multiply $\frac{1}{4}$ by $\frac{\sqrt{L \cdot 2}}{2}$ .

$\frac{44}{7} \cdot \frac{\sqrt{L \cdot 2}}{4 \cdot 2} \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.7.2

Multiply $4$ by $2$ .

$\frac{44}{7} \cdot \frac{\sqrt{L \cdot 2}}{8} \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.8

Cancel the common factor of $4$ .

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Step 3.2.8.1

Factor $4$ out of $44$ .

$\frac{4 (11)}{7} \cdot \frac{\sqrt{L \cdot 2}}{8} \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.8.2

Factor $4$ out of $8$ .

$\frac{4 \cdot 11}{7} \cdot \frac{\sqrt{L \cdot 2}}{4 \cdot 2} \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.8.3

Cancel the common factor.

$\frac{4 \cdot 11}{7} \cdot \frac{\sqrt{L \cdot 2}}{4 \cdot 2} \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.8.4

Rewrite the expression.

$\frac{11}{7} \cdot \frac{\sqrt{L \cdot 2}}{2} \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.9

Multiply $\frac{11}{7}$ by $\frac{\sqrt{L \cdot 2}}{2}$ .

$\frac{11 \sqrt{L \cdot 2}}{7 \cdot 2} \cdot \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.10

Multiply $7$ by $2$ .

$\frac{11 \sqrt{L \cdot 2}}{14} \cdot \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.11

Cancel the common factor of $11$ .

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Step 3.2.11.1

Factor $11$ out of $11 \sqrt{L \cdot 2}$ .

$\frac{11 (\sqrt{L \cdot 2})}{14} \cdot \frac{7}{44} = 6 (\frac{7}{44})$

Step 3.2.11.2

Factor $11$ out of $44$ .

$\frac{11 (\sqrt{L \cdot 2})}{14} \cdot \frac{7}{11 (4)} = 6 (\frac{7}{44})$

Step 3.2.11.3

Cancel the common factor.

$\frac{11 \sqrt{L \cdot 2}}{14} \cdot \frac{7}{11 \cdot 4} = 6 (\frac{7}{44})$

Step 3.2.11.4

Rewrite the expression.

$\frac{\sqrt{L \cdot 2}}{14} \cdot \frac{7}{4} = 6 (\frac{7}{44})$

Step 3.2.12

Cancel the common factor of $7$ .

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Step 3.2.12.1

Factor $7$ out of $14$ .

$\frac{\sqrt{L \cdot 2}}{7 (2)} \cdot \frac{7}{4} = 6 (\frac{7}{44})$

Step 3.2.12.2

Cancel the common factor.

$\frac{\sqrt{L \cdot 2}}{7 \cdot 2} \cdot \frac{7}{4} = 6 (\frac{7}{44})$

Step 3.2.12.3

Rewrite the expression.

$\frac{\sqrt{L \cdot 2}}{2} \cdot \frac{1}{4} = 6 (\frac{7}{44})$

Step 3.2.13

Multiply $\frac{\sqrt{L \cdot 2}}{2}$ by $\frac{1}{4}$ .

$\frac{\sqrt{L \cdot 2}}{2 \cdot 4} = 6 (\frac{7}{44})$

Step 3.2.14

Simplify the expression.

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Step 3.2.14.1

Multiply $2$ by $4$ .

$\frac{\sqrt{L \cdot 2}}{8} = 6 (\frac{7}{44})$

Step 3.2.14.2

Reorder factors in $\frac{\sqrt{L \cdot 2}}{8}$ .

$\frac{\sqrt{2 L}}{8} = 6 (\frac{7}{44})$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $2$ .

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Step 3.3.1.1

Factor $2$ out of $6$ .

$\frac{\sqrt{2 L}}{8} = 2 (3) \frac{7}{44}$

Step 3.3.1.2

Factor $2$ out of $44$ .

$\frac{\sqrt{2 L}}{8} = 2 \cdot 3 \frac{7}{2 \cdot 22}$

Step 3.3.1.3

Cancel the common factor.

$\frac{\sqrt{2 L}}{8} = 2 \cdot 3 \frac{7}{2 \cdot 22}$

Step 3.3.1.4

Rewrite the expression.

$\frac{\sqrt{2 L}}{8} = 3 (\frac{7}{22})$

Step 3.3.2

Combine $3$ and $\frac{7}{22}$ .

$\frac{\sqrt{2 L}}{8} = \frac{3 \cdot 7}{22}$

Step 3.3.3

Multiply $3$ by $7$ .

$\frac{\sqrt{2 L}}{8} = \frac{21}{22}$

Step 4

Multiply both sides by $8$ .

$\frac{\sqrt{2 L}}{8} \cdot 8 = \frac{21}{22} \cdot 8$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $8$ .

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Step 5.1.1.1

Cancel the common factor.

$\frac{\sqrt{2 L}}{8} \cdot 8 = \frac{21}{22} \cdot 8$

Step 5.1.1.2

Rewrite the expression.

$\sqrt{2 L} = \frac{21}{22} \cdot 8$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{21}{22} \cdot 8$ .

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Step 5.2.1.1

Cancel the common factor of $2$ .

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Step 5.2.1.1.1

Factor $2$ out of $22$ .

$\sqrt{2 L} = \frac{21}{2 (11)} \cdot 8$

Step 5.2.1.1.2

Factor $2$ out of $8$ .

$\sqrt{2 L} = \frac{21}{2 \cdot 11} \cdot (2 \cdot 4)$

Step 5.2.1.1.3

Cancel the common factor.

$\sqrt{2 L} = \frac{21}{2 \cdot 11} \cdot (2 \cdot 4)$

Step 5.2.1.1.4

Rewrite the expression.

$\sqrt{2 L} = \frac{21}{11} \cdot 4$

Step 5.2.1.2

Combine $\frac{21}{11}$ and $4$ .

$\sqrt{2 L} = \frac{21 \cdot 4}{11}$

Step 5.2.1.3

Multiply $21$ by $4$ .

$\sqrt{2 L} = \frac{84}{11}$

Step 6

Solve for $L$ .

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Step 6.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{2 L}}^{2} = {(\frac{84}{11})}^{2}$

Step 6.2

Simplify each side of the equation.

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Step 6.2.1

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

${({(2 L)}^{\frac{1}{2}})}^{2} = {(\frac{84}{11})}^{2}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Simplify ${({(2 L)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.2.1.1

Multiply the exponents in ${({(2 L)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.2.1.1.1

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

${(2 L)}^{\frac{1}{2} \cdot 2} = {(\frac{84}{11})}^{2}$

Step 6.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.2.2.1.1.2.1

Cancel the common factor.

${(2 L)}^{\frac{1}{2} \cdot 2} = {(\frac{84}{11})}^{2}$

Step 6.2.2.1.1.2.2

Rewrite the expression.

${(2 L)}^{1} = {(\frac{84}{11})}^{2}$

Step 6.2.2.1.2

Simplify.

$2 L = {(\frac{84}{11})}^{2}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Simplify ${(\frac{84}{11})}^{2}$ .

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Step 6.2.3.1.1

Apply the product rule to $\frac{84}{11}$ .

$2 L = \frac{84^{2}}{11^{2}}$

Step 6.2.3.1.2

Raise $84$ to the power of $2$ .

$2 L = \frac{7056}{11^{2}}$

Step 6.2.3.1.3

Raise $11$ to the power of $2$ .

$2 L = \frac{7056}{121}$

Step 6.3

Divide each term in $2 L = \frac{7056}{121}$ by $2$ and simplify.

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Step 6.3.1

Divide each term in $2 L = \frac{7056}{121}$ by $2$ .

$\frac{2 L}{2} = \frac{\frac{7056}{121}}{2}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $2$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{2 L}{2} = \frac{\frac{7056}{121}}{2}$

Step 6.3.2.1.2

Divide $L$ by $1$ .

$L = \frac{\frac{7056}{121}}{2}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$L = \frac{7056}{121} \cdot \frac{1}{2}$

Step 6.3.3.2

Cancel the common factor of $2$ .

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Step 6.3.3.2.1

Factor $2$ out of $7056$ .

$L = \frac{2 (3528)}{121} \cdot \frac{1}{2}$

Step 6.3.3.2.2

Cancel the common factor.

$L = \frac{2 \cdot 3528}{121} \cdot \frac{1}{2}$

Step 6.3.3.2.3

Rewrite the expression.

$L = \frac{3528}{121}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$L = \frac{3528}{121}$

Decimal Form:

$L = 29.15702479 \dots$

Mixed Number Form:

$L = 29 \frac{19}{121}$