Algebra Examples

$l = 54 \sqrt{d^{3}}$

Step 1

Rewrite the equation as $54 \sqrt{d^{3}} = l$ .

$54 \sqrt{d^{3}} = l$

Step 2

Divide each term in $54 \sqrt{d^{3}} = l$ by $54$ and simplify.

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Step 2.1

Divide each term in $54 \sqrt{d^{3}} = l$ by $54$ .

$\frac{54 \sqrt{d^{3}}}{54} = \frac{l}{54}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $54$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{54 \sqrt{d^{3}}}{54} = \frac{l}{54}$

Step 2.2.1.2

Divide $\sqrt{d^{3}}$ by $1$ .

$\sqrt{d^{3}} = \frac{l}{54}$

Step 2.2.2

Factor out $d^{2}$ .

$\sqrt{d^{2} d} = \frac{l}{54}$

Step 2.2.3

Pull terms out from under the radical.

$d \sqrt{d} = \frac{l}{54}$

Step 3

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

${(d \sqrt{d})}^{2} = {(\frac{l}{54})}^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

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

${(d \cdot d^{\frac{1}{2}})}^{2} = {(\frac{l}{54})}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

Multiply $d$ by $d^{\frac{1}{2}}$ by adding the exponents.

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Step 4.2.1.1.1

Multiply $d$ by $d^{\frac{1}{2}}$ .

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Step 4.2.1.1.1.1

Raise $d$ to the power of $1$ .

${(d^{1} d^{\frac{1}{2}})}^{2} = {(\frac{l}{54})}^{2}$

Step 4.2.1.1.1.2

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

${(d^{1 + \frac{1}{2}})}^{2} = {(\frac{l}{54})}^{2}$

Step 4.2.1.1.2

Write $1$ as a fraction with a common denominator.

${(d^{\frac{2}{2} + \frac{1}{2}})}^{2} = {(\frac{l}{54})}^{2}$

Step 4.2.1.1.3

Combine the numerators over the common denominator.

${(d^{\frac{2 + 1}{2}})}^{2} = {(\frac{l}{54})}^{2}$

Step 4.2.1.1.4

Add $2$ and $1$ .

${(d^{\frac{3}{2}})}^{2} = {(\frac{l}{54})}^{2}$

Step 4.2.1.2

Multiply the exponents in ${(d^{\frac{3}{2}})}^{2}$ .

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Step 4.2.1.2.1

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

$d^{\frac{3}{2} \cdot 2} = {(\frac{l}{54})}^{2}$

Step 4.2.1.2.2

Cancel the common factor of $2$ .

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Step 4.2.1.2.2.1

Cancel the common factor.

$d^{\frac{3}{2} \cdot 2} = {(\frac{l}{54})}^{2}$

Step 4.2.1.2.2.2

Rewrite the expression.

$d^{3} = {(\frac{l}{54})}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(\frac{l}{54})}^{2}$ .

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Step 4.3.1.1

Apply the product rule to $\frac{l}{54}$ .

$d^{3} = \frac{l^{2}}{54^{2}}$

Step 4.3.1.2

Raise $54$ to the power of $2$ .

$d^{3} = \frac{l^{2}}{2916}$

Step 5

Solve for $d$ .

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Step 5.1

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

$d = \sqrt[3]{\frac{l^{2}}{2916}}$

Step 5.2

Simplify $\sqrt[3]{\frac{l^{2}}{2916}}$ .

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Step 5.2.1

Rewrite $\frac{l^{2}}{2916}$ as ${(\frac{1}{9})}^{3} \frac{l^{2}}{4}$ .

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Step 5.2.1.1

Factor the perfect power $1^{3}$ out of $l^{2}$ .

$d = \sqrt[3]{\frac{1^{3} l^{2}}{2916}}$

Step 5.2.1.2

Factor the perfect power $9^{3}$ out of $2916$ .

$d = \sqrt[3]{\frac{1^{3} l^{2}}{9^{3} \cdot 4}}$

Step 5.2.1.3

Rearrange the fraction $\frac{1^{3} l^{2}}{9^{3} \cdot 4}$ .

$d = \sqrt[3]{{(\frac{1}{9})}^{3} \frac{l^{2}}{4}}$

Step 5.2.2

Pull terms out from under the radical.

$d = \frac{1}{9} \sqrt[3]{\frac{l^{2}}{4}}$

Step 5.2.3

Rewrite $\sqrt[3]{\frac{l^{2}}{4}}$ as $\frac{\sqrt[3]{l^{2}}}{\sqrt[3]{4}}$ .

$d = \frac{1}{9} \cdot \frac{\sqrt[3]{l^{2}}}{\sqrt[3]{4}}$

Step 5.2.4

Combine.

$d = \frac{1 \sqrt[3]{l^{2}}}{9 \sqrt[3]{4}}$

Step 5.2.5

Multiply $\sqrt[3]{l^{2}}$ by $1$ .

$d = \frac{\sqrt[3]{l^{2}}}{9 \sqrt[3]{4}}$

Step 5.2.6

Multiply $\frac{\sqrt[3]{l^{2}}}{9 \sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$d = \frac{\sqrt[3]{l^{2}}}{9 \sqrt[3]{4}} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$

Step 5.2.7

Combine and simplify the denominator.

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Step 5.2.7.1

Multiply $\frac{\sqrt[3]{l^{2}}}{9 \sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$d = \frac{\sqrt[3]{l^{2}} {\sqrt[3]{4}}^{2}}{9 \sqrt[3]{4} {\sqrt[3]{4}}^{2}}$

Step 5.2.7.2

Move $\sqrt[3]{4}$ .

$d = \frac{\sqrt[3]{l^{2}} {\sqrt[3]{4}}^{2}}{9 (\sqrt[3]{4} {\sqrt[3]{4}}^{2})}$

Step 5.2.7.3

Raise $\sqrt[3]{4}$ to the power of $1$ .

$d = \frac{\sqrt[3]{l^{2}} {\sqrt[3]{4}}^{2}}{9 ({\sqrt[3]{4}}^{1} {\sqrt[3]{4}}^{2})}$

Step 5.2.7.4

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

$d = \frac{\sqrt[3]{l^{2}} {\sqrt[3]{4}}^{2}}{9 {\sqrt[3]{4}}^{1 + 2}}$

Step 5.2.7.5

Add $1$ and $2$ .

$d = \frac{\sqrt[3]{l^{2}} {\sqrt[3]{4}}^{2}}{9 {\sqrt[3]{4}}^{3}}$

Step 5.2.7.6

Rewrite ${\sqrt[3]{4}}^{3}$ as $4$ .

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Step 5.2.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$d = \frac{\sqrt[3]{l^{2}} {\sqrt[3]{4}}^{2}}{9 {(4^{\frac{1}{3}})}^{3}}$

Step 5.2.7.6.2

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

$d = \frac{\sqrt[3]{l^{2}} {\sqrt[3]{4}}^{2}}{9 \cdot 4^{\frac{1}{3} \cdot 3}}$

Step 5.2.7.6.3

Combine $\frac{1}{3}$ and $3$ .

$d = \frac{\sqrt[3]{l^{2}} {\sqrt[3]{4}}^{2}}{9 \cdot 4^{\frac{3}{3}}}$

Step 5.2.7.6.4

Cancel the common factor of $3$ .

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Step 5.2.7.6.4.1

Cancel the common factor.

$d = \frac{\sqrt[3]{l^{2}} {\sqrt[3]{4}}^{2}}{9 \cdot 4^{\frac{3}{3}}}$

Step 5.2.7.6.4.2

Rewrite the expression.

$d = \frac{\sqrt[3]{l^{2}} {\sqrt[3]{4}}^{2}}{9 \cdot 4^{1}}$

Step 5.2.7.6.5

Evaluate the exponent.

$d = \frac{\sqrt[3]{l^{2}} {\sqrt[3]{4}}^{2}}{9 \cdot 4}$

Step 5.2.8

Simplify the numerator.

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Step 5.2.8.1

Rewrite ${\sqrt[3]{4}}^{2}$ as $\sqrt[3]{4^{2}}$ .

$d = \frac{\sqrt[3]{l^{2}} \sqrt[3]{4^{2}}}{9 \cdot 4}$

Step 5.2.8.2

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

$d = \frac{\sqrt[3]{l^{2}} \sqrt[3]{16}}{9 \cdot 4}$

Step 5.2.8.3

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 5.2.8.3.1

Factor $8$ out of $16$ .

$d = \frac{\sqrt[3]{l^{2}} \sqrt[3]{8 (2)}}{9 \cdot 4}$

Step 5.2.8.3.2

Rewrite $8$ as $2^{3}$ .

$d = \frac{\sqrt[3]{l^{2}} \sqrt[3]{2^{3} \cdot 2}}{9 \cdot 4}$

Step 5.2.8.4

Pull terms out from under the radical.

$d = \frac{\sqrt[3]{l^{2}} \cdot 2 \sqrt[3]{2}}{9 \cdot 4}$

Step 5.2.8.5

Combine using the product rule for radicals.

$d = \frac{2 \sqrt[3]{l^{2} \cdot 2}}{9 \cdot 4}$

Step 5.2.9

Reduce the expression by cancelling the common factors.

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Step 5.2.9.1

Multiply $9$ by $4$ .

$d = \frac{2 \sqrt[3]{l^{2} \cdot 2}}{36}$

Step 5.2.9.2

Cancel the common factor of $2$ and $36$ .

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Step 5.2.9.2.1

Factor $2$ out of $2 \sqrt[3]{l^{2} \cdot 2}$ .

$d = \frac{2 (\sqrt[3]{l^{2} \cdot 2})}{36}$

Step 5.2.9.2.2

Cancel the common factors.

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Step 5.2.9.2.2.1

Factor $2$ out of $36$ .

$d = \frac{2 \sqrt[3]{l^{2} \cdot 2}}{2 \cdot 18}$

Step 5.2.9.2.2.2

Cancel the common factor.

$d = \frac{2 \sqrt[3]{l^{2} \cdot 2}}{2 \cdot 18}$

Step 5.2.9.2.2.3

Rewrite the expression.

$d = \frac{\sqrt[3]{l^{2} \cdot 2}}{18}$

Step 5.2.9.3

Reorder factors in $\frac{\sqrt[3]{l^{2} \cdot 2}}{18}$ .

$d = \frac{\sqrt[3]{2 l^{2}}}{18}$