Algebra Examples

${(x^{2} y^{3})}^{\frac{1}{2}} {(x^{2} y^{3})}^{\frac{1}{3}} = x^{\frac{a}{3}} y^{\frac{a}{2}}$

Step 1

Rewrite the equation as $x^{\frac{a}{3}} y^{\frac{a}{2}} = {(x^{2} y^{3})}^{\frac{1}{2}} {(x^{2} y^{3})}^{\frac{1}{3}}$ .

$x^{\frac{a}{3}} y^{\frac{a}{2}} = {(x^{2} y^{3})}^{\frac{1}{2}} {(x^{2} y^{3})}^{\frac{1}{3}}$

Step 2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (x^{\frac{a}{3}} y^{\frac{a}{2}}) = \ln ({(x^{2} y^{3})}^{\frac{1}{2}} {(x^{2} y^{3})}^{\frac{1}{3}})$

Step 3

Expand the left side.

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Step 3.1

Rewrite $\ln (x^{\frac{a}{3}} y^{\frac{a}{2}})$ as $\ln (x^{\frac{a}{3}}) + \ln (y^{\frac{a}{2}})$ .

$\ln (x^{\frac{a}{3}}) + \ln (y^{\frac{a}{2}}) = \ln ({(x^{2} y^{3})}^{\frac{1}{2}} {(x^{2} y^{3})}^{\frac{1}{3}})$

Step 3.2

Expand $\ln (x^{\frac{a}{3}})$ by moving $\frac{a}{3}$ outside the logarithm.

$\frac{a}{3} \ln (x) + \ln (y^{\frac{a}{2}}) = \ln ({(x^{2} y^{3})}^{\frac{1}{2}} {(x^{2} y^{3})}^{\frac{1}{3}})$

Step 3.3

Expand $\ln (y^{\frac{a}{2}})$ by moving $\frac{a}{2}$ outside the logarithm.

$\frac{a}{3} \ln (x) + \frac{a}{2} \ln (y) = \ln ({(x^{2} y^{3})}^{\frac{1}{2}} {(x^{2} y^{3})}^{\frac{1}{3}})$

Step 3.4

Combine $\frac{a}{3}$ and $\ln (x)$ .

$\frac{a \ln (x)}{3} + \frac{a}{2} \ln (y) = \ln ({(x^{2} y^{3})}^{\frac{1}{2}} {(x^{2} y^{3})}^{\frac{1}{3}})$

Step 3.5

Combine $\frac{a}{2}$ and $\ln (y)$ .

$\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln ({(x^{2} y^{3})}^{\frac{1}{2}} {(x^{2} y^{3})}^{\frac{1}{3}})$

Step 4

Simplify the right side.

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Step 4.1

Simplify $\ln ({(x^{2} y^{3})}^{\frac{1}{2}} {(x^{2} y^{3})}^{\frac{1}{3}})$ .

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Step 4.1.1

Multiply ${(x^{2} y^{3})}^{\frac{1}{2}}$ by ${(x^{2} y^{3})}^{\frac{1}{3}}$ by adding the exponents.

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Step 4.1.1.1

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

$\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln ({(x^{2} y^{3})}^{\frac{1}{2} + \frac{1}{3}})$

Step 4.1.1.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln ({(x^{2} y^{3})}^{\frac{1}{2} \cdot \frac{3}{3} + \frac{1}{3}})$

Step 4.1.1.3

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln ({(x^{2} y^{3})}^{\frac{1}{2} \cdot \frac{3}{3} + \frac{1}{3} \cdot \frac{2}{2}})$

Step 4.1.1.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.1.4.1

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln ({(x^{2} y^{3})}^{\frac{3}{2 \cdot 3} + \frac{1}{3} \cdot \frac{2}{2}})$

Step 4.1.1.4.2

Multiply $2$ by $3$ .

$\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln ({(x^{2} y^{3})}^{\frac{3}{6} + \frac{1}{3} \cdot \frac{2}{2}})$

Step 4.1.1.4.3

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln ({(x^{2} y^{3})}^{\frac{3}{6} + \frac{2}{3 \cdot 2}})$

Step 4.1.1.4.4

Multiply $3$ by $2$ .

$\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln ({(x^{2} y^{3})}^{\frac{3}{6} + \frac{2}{6}})$

Step 4.1.1.5

Combine the numerators over the common denominator.

$\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln ({(x^{2} y^{3})}^{\frac{3 + 2}{6}})$

Step 4.1.1.6

Add $3$ and $2$ .

$\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln ({(x^{2} y^{3})}^{\frac{5}{6}})$

Step 4.1.2

Apply the product rule to $x^{2} y^{3}$ .

$\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln ({(x^{2})}^{\frac{5}{6}} {(y^{3})}^{\frac{5}{6}})$

Step 4.1.3

Multiply the exponents in ${(x^{2})}^{\frac{5}{6}}$ .

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Step 4.1.3.1

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

$\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln (x^{2 (\frac{5}{6})} {(y^{3})}^{\frac{5}{6}})$

Step 4.1.3.2

Cancel the common factor of $2$ .

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Step 4.1.3.2.1

Factor $2$ out of $6$ .

$\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln (x^{2 \frac{5}{2 (3)}} {(y^{3})}^{\frac{5}{6}})$

Step 4.1.3.2.2

Cancel the common factor.

$\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln (x^{2 \frac{5}{2 \cdot 3}} {(y^{3})}^{\frac{5}{6}})$

Step 4.1.3.2.3

Rewrite the expression.

$\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln (x^{\frac{5}{3}} {(y^{3})}^{\frac{5}{6}})$

Step 4.1.4

Multiply the exponents in ${(y^{3})}^{\frac{5}{6}}$ .

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Step 4.1.4.1

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

$\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln (x^{\frac{5}{3}} y^{3 (\frac{5}{6})})$

Step 4.1.4.2

Cancel the common factor of $3$ .

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Step 4.1.4.2.1

Factor $3$ out of $6$ .

$\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln (x^{\frac{5}{3}} y^{3 \frac{5}{3 (2)}})$

Step 4.1.4.2.2

Cancel the common factor.

$\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln (x^{\frac{5}{3}} y^{3 \frac{5}{3 \cdot 2}})$

Step 4.1.4.2.3

Rewrite the expression.

$\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln (x^{\frac{5}{3}} y^{\frac{5}{2}})$

Step 5

Multiply each term in $\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln (x^{\frac{5}{3}} y^{\frac{5}{2}})$ by $6$ to eliminate the fractions.

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Step 5.1

Multiply each term in $\frac{a \ln (x)}{3} + \frac{a \ln (y)}{2} = \ln (x^{\frac{5}{3}} y^{\frac{5}{2}})$ by $6$ .

$\frac{a \ln (x)}{3} \cdot 6 + \frac{a \ln (y)}{2} \cdot 6 = \ln (x^{\frac{5}{3}} y^{\frac{5}{2}}) \cdot 6$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Cancel the common factor of $3$ .

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Step 5.2.1.1.1

Factor $3$ out of $6$ .

$\frac{a \ln (x)}{3} \cdot (3 (2)) + \frac{a \ln (y)}{2} \cdot 6 = \ln (x^{\frac{5}{3}} y^{\frac{5}{2}}) \cdot 6$

Step 5.2.1.1.2

Cancel the common factor.

$\frac{a \ln (x)}{3} \cdot (3 \cdot 2) + \frac{a \ln (y)}{2} \cdot 6 = \ln (x^{\frac{5}{3}} y^{\frac{5}{2}}) \cdot 6$

Step 5.2.1.1.3

Rewrite the expression.

$a \ln (x) \cdot 2 + \frac{a \ln (y)}{2} \cdot 6 = \ln (x^{\frac{5}{3}} y^{\frac{5}{2}}) \cdot 6$

Step 5.2.1.2

Move $2$ to the left of $a \ln (x)$ .

$2 \cdot (a \ln (x)) + \frac{a \ln (y)}{2} \cdot 6 = \ln (x^{\frac{5}{3}} y^{\frac{5}{2}}) \cdot 6$

Step 5.2.1.3

Cancel the common factor of $2$ .

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Step 5.2.1.3.1

Factor $2$ out of $6$ .

$2 a \ln (x) + \frac{a \ln (y)}{2} \cdot (2 (3)) = \ln (x^{\frac{5}{3}} y^{\frac{5}{2}}) \cdot 6$

Step 5.2.1.3.2

Cancel the common factor.

$2 a \ln (x) + \frac{a \ln (y)}{2} \cdot (2 \cdot 3) = \ln (x^{\frac{5}{3}} y^{\frac{5}{2}}) \cdot 6$

Step 5.2.1.3.3

Rewrite the expression.

$2 a \ln (x) + a \ln (y) \cdot 3 = \ln (x^{\frac{5}{3}} y^{\frac{5}{2}}) \cdot 6$

Step 5.2.1.4

Move $3$ to the left of $a \ln (y)$ .

$2 a \ln (x) + 3 a \ln (y) = \ln (x^{\frac{5}{3}} y^{\frac{5}{2}}) \cdot 6$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move $6$ to the left of $\ln (x^{\frac{5}{3}} y^{\frac{5}{2}})$ .

$2 a \ln (x) + 3 a \ln (y) = 6 \ln (x^{\frac{5}{3}} y^{\frac{5}{2}})$

Step 6

Move all the terms containing a logarithm to the left side of the equation.

$2 a \ln (x) + 3 a \ln (y) - 6 \ln (x^{\frac{5}{3}} y^{\frac{5}{2}}) = 0$

Step 7

Add $6 \ln (x^{\frac{5}{3}} y^{\frac{5}{2}})$ to both sides of the equation.

$2 a \ln (x) + 3 a \ln (y) = 6 \ln (x^{\frac{5}{3}} y^{\frac{5}{2}})$

Step 8

Factor $a$ out of $2 a \ln (x) + 3 a \ln (y)$ .

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Step 8.1

Factor $a$ out of $2 a \ln (x)$ .

$a (2 \ln (x)) + 3 a \ln (y) = 6 \ln (x^{\frac{5}{3}} y^{\frac{5}{2}})$

Step 8.2

Factor $a$ out of $3 a \ln (y)$ .

$a (2 \ln (x)) + a (3 \ln (y)) = 6 \ln (x^{\frac{5}{3}} y^{\frac{5}{2}})$

Step 8.3

Factor $a$ out of $a (2 \ln (x)) + a (3 \ln (y))$ .

$a (2 \ln (x) + 3 \ln (y)) = 6 \ln (x^{\frac{5}{3}} y^{\frac{5}{2}})$

Step 9

Divide each term in $a (2 \ln (x) + 3 \ln (y)) = 6 \ln (x^{\frac{5}{3}} y^{\frac{5}{2}})$ by $2 \ln (x) + 3 \ln (y)$ and simplify.

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Step 9.1

Divide each term in $a (2 \ln (x) + 3 \ln (y)) = 6 \ln (x^{\frac{5}{3}} y^{\frac{5}{2}})$ by $2 \ln (x) + 3 \ln (y)$ .

$\frac{a (2 \ln (x) + 3 \ln (y))}{2 \ln (x) + 3 \ln (y)} = \frac{6 \ln (x^{\frac{5}{3}} y^{\frac{5}{2}})}{2 \ln (x) + 3 \ln (y)}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $2 \ln (x) + 3 \ln (y)$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{a (2 \ln (x) + 3 \ln (y))}{2 \ln (x) + 3 \ln (y)} = \frac{6 \ln (x^{\frac{5}{3}} y^{\frac{5}{2}})}{2 \ln (x) + 3 \ln (y)}$

Step 9.2.1.2

Divide $a$ by $1$ .

$a = \frac{6 \ln (x^{\frac{5}{3}} y^{\frac{5}{2}})}{2 \ln (x) + 3 \ln (y)}$