Basic Math Examples

$\log (\sqrt[3]{A}) = \log (\sqrt{A})$

Step 1

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$\sqrt[3]{A} = \sqrt{A}$

Step 2

Solve for $A$ .

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

${\sqrt[3]{A}}^{3} = {\sqrt{A}}^{3}$

Simplify each side of the equation.

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

${(A^{\frac{1}{3}})}^{3} = {\sqrt{A}}^{3}$

Simplify the left side.

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Simplify ${(A^{\frac{1}{3}})}^{3}$ .

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Multiply the exponents in ${(A^{\frac{1}{3}})}^{3}$ .

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

$A^{\frac{1}{3} \cdot 3} = {\sqrt{A}}^{3}$

Cancel the common factor of $3$ .

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Cancel the common factor.

$A^{\frac{1}{3} \cdot 3} = {\sqrt{A}}^{3}$

Rewrite the expression.

$A^{1} = {\sqrt{A}}^{3}$

Simplify.

$A = {\sqrt{A}}^{3}$

Simplify the right side.

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Simplify ${\sqrt{A}}^{3}$ .

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Rewrite ${\sqrt{A}}^{3}$ as ${(A^{3})}^{\frac{1}{2}}$ .

$A = \sqrt{A^{3}}$

Factor out $A^{2}$ .

$A = \sqrt{A^{2} A}$

Pull terms out from under the radical.

$A = A \sqrt{A}$

Rewrite the equation as $A \sqrt{A} = A$ .

$A \sqrt{A} = A$

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

${(A \sqrt{A})}^{2} = A^{2}$

Simplify each side of the equation.

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

${(A \cdot A^{\frac{1}{2}})}^{2} = A^{2}$

Simplify the left side.

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

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

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

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

${(A^{1} A^{\frac{1}{2}})}^{2} = A^{2}$

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

${(A^{1 + \frac{1}{2}})}^{2} = A^{2}$

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

${(A^{\frac{2}{2} + \frac{1}{2}})}^{2} = A^{2}$

Combine the numerators over the common denominator.

${(A^{\frac{2 + 1}{2}})}^{2} = A^{2}$

Add $2$ and $1$ .

${(A^{\frac{3}{2}})}^{2} = A^{2}$

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

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

$A^{\frac{3}{2} \cdot 2} = A^{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$A^{\frac{3}{2} \cdot 2} = A^{2}$

Rewrite the expression.

$A^{3} = A^{2}$

Solve for $A$ .

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Subtract $A^{2}$ from both sides of the equation.

$A^{3} - A^{2} = 0$

Factor $A^{2}$ out of $A^{3} - A^{2}$ .

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Factor $A^{2}$ out of $A^{3}$ .

$A^{2} A - A^{2} = 0$

Factor $A^{2}$ out of $- A^{2}$ .

$A^{2} A + A^{2} \cdot - 1 = 0$

Factor $A^{2}$ out of $A^{2} A + A^{2} \cdot - 1$ .

$A^{2} (A - 1) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$A^{2} = 0$

$A - 1 = 0$

Set $A^{2}$ equal to $0$ and solve for $A$ .

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Set $A^{2}$ equal to $0$ .

$A^{2} = 0$

Solve $A^{2} = 0$ for $A$ .

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

$A = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

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Rewrite $0$ as $0^{2}$ .

$A = \pm \sqrt{0^{2}}$

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

$A = \pm 0$

Plus or minus $0$ is $0$ .

$A = 0$

Set $A - 1$ equal to $0$ and solve for $A$ .

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Set $A - 1$ equal to $0$ .

$A - 1 = 0$

Add $1$ to both sides of the equation.

$A = 1$

The final solution is all the values that make $A^{2} (A - 1) = 0$ true.

$A = 0, 1$

Step 3

Exclude the solutions that do not make $\log (\sqrt[3]{A}) = \log (\sqrt{A})$ true.

$A = 1$