Algebra Examples

$\sqrt{\frac{126 x y^{5}}{32 x^{3}}} = \sqrt{\frac{63 y^{5}}{a x^{b}}}$

Step 1

Rewrite the equation as $\sqrt{\frac{63 y^{5}}{a x^{b}}} = \sqrt{\frac{126 x y^{5}}{32 x^{3}}}$ .

$\sqrt{\frac{63 y^{5}}{a x^{b}}} = \sqrt{\frac{126 x y^{5}}{32 x^{3}}}$

Step 2

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

${\sqrt{\frac{63 y^{5}}{a x^{b}}}}^{2} = {\sqrt{\frac{126 x y^{5}}{32 x^{3}}}}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{63 y^{5}}{a x^{b}}}$ as ${(\frac{63 y^{5}}{a x^{b}})}^{\frac{1}{2}}$ .

${({(\frac{63 y^{5}}{a x^{b}})}^{\frac{1}{2}})}^{2} = {\sqrt{\frac{126 x y^{5}}{32 x^{3}}}}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(\frac{63 y^{5}}{a x^{b}})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(\frac{63 y^{5}}{a x^{b}})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(\frac{63 y^{5}}{a x^{b}})}^{\frac{1}{2} \cdot 2} = {\sqrt{\frac{126 x y^{5}}{32 x^{3}}}}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(\frac{63 y^{5}}{a x^{b}})}^{\frac{1}{2} \cdot 2} = {\sqrt{\frac{126 x y^{5}}{32 x^{3}}}}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(\frac{63 y^{5}}{a x^{b}})}^{1} = {\sqrt{\frac{126 x y^{5}}{32 x^{3}}}}^{2}$

Step 3.2.1.2

Simplify.

$\frac{63 y^{5}}{a x^{b}} = {\sqrt{\frac{126 x y^{5}}{32 x^{3}}}}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${\sqrt{\frac{126 x y^{5}}{32 x^{3}}}}^{2}$ .

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Step 3.3.1.1

Reduce the expression $\frac{126 x y^{5}}{32 x^{3}}$ by cancelling the common factors.

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Step 3.3.1.1.1

Factor $2$ out of $126 x y^{5}$ .

$\frac{63 y^{5}}{a x^{b}} = {\sqrt{\frac{2 (63 x y^{5})}{32 x^{3}}}}^{2}$

Step 3.3.1.1.2

Factor $2$ out of $32 x^{3}$ .

$\frac{63 y^{5}}{a x^{b}} = {\sqrt{\frac{2 (63 x y^{5})}{2 (16 x^{3})}}}^{2}$

Step 3.3.1.1.3

Cancel the common factor.

$\frac{63 y^{5}}{a x^{b}} = {\sqrt{\frac{2 (63 x y^{5})}{2 (16 x^{3})}}}^{2}$

Step 3.3.1.1.4

Rewrite the expression.

$\frac{63 y^{5}}{a x^{b}} = {\sqrt{\frac{63 x y^{5}}{16 x^{3}}}}^{2}$

Step 3.3.1.2

Cancel the common factor of $x$ and $x^{3}$ .

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Step 3.3.1.2.1

Factor $x$ out of $63 x y^{5}$ .

$\frac{63 y^{5}}{a x^{b}} = {\sqrt{\frac{x (63 y^{5})}{16 x^{3}}}}^{2}$

Step 3.3.1.2.2

Cancel the common factors.

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Step 3.3.1.2.2.1

Factor $x$ out of $16 x^{3}$ .

$\frac{63 y^{5}}{a x^{b}} = {\sqrt{\frac{x (63 y^{5})}{x (16 x^{2})}}}^{2}$

Step 3.3.1.2.2.2

Cancel the common factor.

$\frac{63 y^{5}}{a x^{b}} = {\sqrt{\frac{x (63 y^{5})}{x (16 x^{2})}}}^{2}$

Step 3.3.1.2.2.3

Rewrite the expression.

$\frac{63 y^{5}}{a x^{b}} = {\sqrt{\frac{63 y^{5}}{16 x^{2}}}}^{2}$

Step 3.3.1.3

Rewrite $\frac{63 y^{5}}{16 x^{2}}$ as ${(\frac{3 y^{2}}{4 x})}^{2} (7 y)$ .

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Step 3.3.1.3.1

Factor the perfect power ${(3 y^{2})}^{2}$ out of $63 y^{5}$ .

$\frac{63 y^{5}}{a x^{b}} = {\sqrt{\frac{{(3 y^{2})}^{2} \cdot (7 y)}{16 x^{2}}}}^{2}$

Step 3.3.1.3.2

Factor the perfect power ${(4 x)}^{2}$ out of $16 x^{2}$ .

$\frac{63 y^{5}}{a x^{b}} = {\sqrt{\frac{{(3 y^{2})}^{2} \cdot (7 y)}{{(4 x)}^{2} \cdot 1}}}^{2}$

Step 3.3.1.3.3

Rearrange the fraction $\frac{{(3 y^{2})}^{2} \cdot (7 y)}{{(4 x)}^{2} \cdot 1}$ .

$\frac{63 y^{5}}{a x^{b}} = {\sqrt{{(\frac{3 y^{2}}{4 x})}^{2} (7 y)}}^{2}$

Step 3.3.1.4

Pull terms out from under the radical.

$\frac{63 y^{5}}{a x^{b}} = {(\frac{3 y^{2}}{4 x} \sqrt{7 y})}^{2}$

Step 3.3.1.5

Combine $\frac{3 y^{2}}{4 x}$ and $\sqrt{7 y}$ .

$\frac{63 y^{5}}{a x^{b}} = {(\frac{3 y^{2} \sqrt{7 y}}{4 x})}^{2}$

Step 3.3.1.6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.3.1.6.1

Apply the product rule to $\frac{3 y^{2} \sqrt{7 y}}{4 x}$ .

$\frac{63 y^{5}}{a x^{b}} = \frac{{(3 y^{2} \sqrt{7 y})}^{2}}{{(4 x)}^{2}}$

Step 3.3.1.6.2

Apply the product rule to $3 y^{2} \sqrt{7 y}$ .

$\frac{63 y^{5}}{a x^{b}} = \frac{{(3 y^{2})}^{2} {\sqrt{7 y}}^{2}}{{(4 x)}^{2}}$

Step 3.3.1.6.3

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

$\frac{63 y^{5}}{a x^{b}} = \frac{3^{2} {(y^{2})}^{2} {\sqrt{7 y}}^{2}}{{(4 x)}^{2}}$

Step 3.3.1.6.4

Apply the product rule to $4 x$ .

$\frac{63 y^{5}}{a x^{b}} = \frac{3^{2} {(y^{2})}^{2} {\sqrt{7 y}}^{2}}{4^{2} x^{2}}$

Step 3.3.1.7

Simplify the numerator.

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Step 3.3.1.7.1

Raise $3$ to the power of $2$ .

$\frac{63 y^{5}}{a x^{b}} = \frac{9 {(y^{2})}^{2} {\sqrt{7 y}}^{2}}{4^{2} x^{2}}$

Step 3.3.1.7.2

Multiply the exponents in ${(y^{2})}^{2}$ .

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Step 3.3.1.7.2.1

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

$\frac{63 y^{5}}{a x^{b}} = \frac{9 y^{2 \cdot 2} {\sqrt{7 y}}^{2}}{4^{2} x^{2}}$

Step 3.3.1.7.2.2

Multiply $2$ by $2$ .

$\frac{63 y^{5}}{a x^{b}} = \frac{9 y^{4} {\sqrt{7 y}}^{2}}{4^{2} x^{2}}$

Step 3.3.1.7.3

Rewrite ${\sqrt{7 y}}^{2}$ as $7 y$ .

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Step 3.3.1.7.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 y}$ as ${(7 y)}^{\frac{1}{2}}$ .

$\frac{63 y^{5}}{a x^{b}} = \frac{9 y^{4} {({(7 y)}^{\frac{1}{2}})}^{2}}{4^{2} x^{2}}$

Step 3.3.1.7.3.2

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

$\frac{63 y^{5}}{a x^{b}} = \frac{9 y^{4} {(7 y)}^{\frac{1}{2} \cdot 2}}{4^{2} x^{2}}$

Step 3.3.1.7.3.3

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

$\frac{63 y^{5}}{a x^{b}} = \frac{9 y^{4} {(7 y)}^{\frac{2}{2}}}{4^{2} x^{2}}$

Step 3.3.1.7.3.4

Cancel the common factor of $2$ .

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Step 3.3.1.7.3.4.1

Cancel the common factor.

$\frac{63 y^{5}}{a x^{b}} = \frac{9 y^{4} {(7 y)}^{\frac{2}{2}}}{4^{2} x^{2}}$

Step 3.3.1.7.3.4.2

Rewrite the expression.

$\frac{63 y^{5}}{a x^{b}} = \frac{9 y^{4} {(7 y)}^{1}}{4^{2} x^{2}}$

Step 3.3.1.7.3.5

Simplify.

$\frac{63 y^{5}}{a x^{b}} = \frac{9 y^{4} (7 y)}{4^{2} x^{2}}$

$\frac{63 y^{5}}{a x^{b}} = \frac{9 y^{4} \cdot 7 y}{4^{2} x^{2}}$

Step 3.3.1.7.4

Combine exponents.

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Step 3.3.1.7.4.1

Multiply $7$ by $9$ .

$\frac{63 y^{5}}{a x^{b}} = \frac{63 y^{4} y}{4^{2} x^{2}}$

Step 3.3.1.7.4.2

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

$\frac{63 y^{5}}{a x^{b}} = \frac{63 (y^{1} y^{4})}{4^{2} x^{2}}$

Step 3.3.1.7.4.3

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

$\frac{63 y^{5}}{a x^{b}} = \frac{63 y^{1 + 4}}{4^{2} x^{2}}$

Step 3.3.1.7.4.4

Add $1$ and $4$ .

$\frac{63 y^{5}}{a x^{b}} = \frac{63 y^{5}}{4^{2} x^{2}}$

Step 3.3.1.8

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

$\frac{63 y^{5}}{a x^{b}} = \frac{63 y^{5}}{16 x^{2}}$

Step 4

Solve for $b$ .

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Step 4.1

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

$\ln (\frac{63 y^{5}}{a x^{b}}) = \ln (\frac{63 y^{5}}{16 x^{2}})$

Step 4.2

Expand the left side.

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Step 4.2.1

Rewrite $\ln (\frac{63 y^{5}}{a x^{b}})$ as $\ln (63 y^{5}) - \ln (a x^{b})$ .

$\ln (63 y^{5}) - \ln (a x^{b}) = \ln (\frac{63 y^{5}}{16 x^{2}})$

Step 4.2.2

Rewrite $\ln (63 y^{5})$ as $\ln (63) + \ln (y^{5})$ .

$\ln (63) + \ln (y^{5}) - \ln (a x^{b}) = \ln (\frac{63 y^{5}}{16 x^{2}})$

Step 4.2.3

Expand $\ln (y^{5})$ by moving $5$ outside the logarithm.

$\ln (63) + 5 \ln (y) - \ln (a x^{b}) = \ln (\frac{63 y^{5}}{16 x^{2}})$

Step 4.2.4

Rewrite $\ln (a x^{b})$ as $\ln (a) + \ln (x^{b})$ .

$\ln (63) + 5 \ln (y) - (\ln (a) + \ln (x^{b})) = \ln (\frac{63 y^{5}}{16 x^{2}})$

Step 4.2.5

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

$\ln (63) + 5 \ln (y) - (\ln (a) + b \ln (x)) = \ln (\frac{63 y^{5}}{16 x^{2}})$

Step 4.3

Simplify the left side.

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Step 4.3.1

Simplify $\ln (63) + 5 \ln (y) - (\ln (a) + b \ln (x))$ .

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Step 4.3.1.1

Apply the distributive property.

$\ln (63) + 5 \ln (y) - \ln (a) - b \ln (x) = \ln (\frac{63 y^{5}}{16 x^{2}})$

Step 4.3.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$5 \ln (y) + \ln (\frac{63}{a}) - b \ln (x) = \ln (\frac{63 y^{5}}{16 x^{2}})$

Step 4.4

Simplify the left side.

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Step 4.4.1

Move $\ln (\frac{63}{a})$ .

$5 \ln (y) - b \ln (x) + \ln (\frac{63}{a}) = \ln (\frac{63 y^{5}}{16 x^{2}})$

Step 4.4.2

Reorder $5 \ln (y)$ and $- b \ln (x)$ .

$- b \ln (x) + 5 \ln (y) + \ln (\frac{63}{a}) = \ln (\frac{63 y^{5}}{16 x^{2}})$

Step 4.5

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

$- b \ln (x) + 5 \ln (y) + \ln (\frac{63}{a}) - \ln (\frac{63 y^{5}}{16 x^{2}}) = 0$

Step 4.6

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$- b \ln (x) + 5 \ln (y) + \ln (\frac{\frac{63}{a}}{\frac{63 y^{5}}{16 x^{2}}}) = 0$

Step 4.7

Simplify each term.

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Step 4.7.1

Multiply the numerator by the reciprocal of the denominator.

$- b \ln (x) + 5 \ln (y) + \ln (\frac{63}{a} \cdot \frac{16 x^{2}}{63 y^{5}}) = 0$

Step 4.7.2

Combine.

$- b \ln (x) + 5 \ln (y) + \ln (\frac{63 (16 x^{2})}{a (63 y^{5})}) = 0$

Step 4.7.3

Cancel the common factor of $63$ .

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Step 4.7.3.1

Cancel the common factor.

$- b \ln (x) + 5 \ln (y) + \ln (\frac{63 (16 x^{2})}{a (63 y^{5})}) = 0$

Step 4.7.3.2

Rewrite the expression.

$- b \ln (x) + 5 \ln (y) + \ln (\frac{16 x^{2}}{a (y^{5})}) = 0$

$- b \ln (x) + 5 \ln (y) + \ln (\frac{16 x^{2}}{a y^{5}}) = 0$

Step 4.8

Move all terms not containing $b$ to the right side of the equation.

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Step 4.8.1

Subtract $5 \ln (y)$ from both sides of the equation.

$- b \ln (x) + \ln (\frac{16 x^{2}}{a y^{5}}) = - 5 \ln (y)$

Step 4.8.2

Subtract $\ln (\frac{16 x^{2}}{a y^{5}})$ from both sides of the equation.

$- b \ln (x) = - 5 \ln (y) - \ln (\frac{16 x^{2}}{a y^{5}})$

Step 4.9

Divide each term in $- b \ln (x) = - 5 \ln (y) - \ln (\frac{16 x^{2}}{a y^{5}})$ by $- \ln (x)$ and simplify.

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Step 4.9.1

Divide each term in $- b \ln (x) = - 5 \ln (y) - \ln (\frac{16 x^{2}}{a y^{5}})$ by $- \ln (x)$ .

$\frac{- b \ln (x)}{- \ln (x)} = \frac{- 5 \ln (y)}{- \ln (x)} + \frac{- \ln (\frac{16 x^{2}}{a y^{5}})}{- \ln (x)}$

Step 4.9.2

Simplify the left side.

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Step 4.9.2.1

Dividing two negative values results in a positive value.

$\frac{b \ln (x)}{\ln (x)} = \frac{- 5 \ln (y)}{- \ln (x)} + \frac{- \ln (\frac{16 x^{2}}{a y^{5}})}{- \ln (x)}$

Step 4.9.2.2

Cancel the common factor of $\ln (x)$ .

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Step 4.9.2.2.1

Cancel the common factor.

$\frac{b \ln (x)}{\ln (x)} = \frac{- 5 \ln (y)}{- \ln (x)} + \frac{- \ln (\frac{16 x^{2}}{a y^{5}})}{- \ln (x)}$

Step 4.9.2.2.2

Divide $b$ by $1$ .

$b = \frac{- 5 \ln (y)}{- \ln (x)} + \frac{- \ln (\frac{16 x^{2}}{a y^{5}})}{- \ln (x)}$

Step 4.9.3

Simplify the right side.

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Step 4.9.3.1

Simplify each term.

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Step 4.9.3.1.1

Dividing two negative values results in a positive value.

$b = \frac{5 \ln (y)}{\ln (x)} + \frac{- \ln (\frac{16 x^{2}}{a y^{5}})}{- \ln (x)}$

Step 4.9.3.1.2

Dividing two negative values results in a positive value.

$b = \frac{5 \ln (y)}{\ln (x)} + \frac{\ln (\frac{16 x^{2}}{a y^{5}})}{\ln (x)}$