Basic Math Examples

$\frac{\sqrt[3]{16^{\frac{z}{2}}}}{2 \cdot \sqrt[6]{64^{- 3}}} = 4$

Step 1

Multiply both sides of the equation by $2 \cdot \sqrt[6]{64^{- 3}}$ .

$2 \cdot \sqrt[6]{64^{- 3}} \frac{\sqrt[3]{16^{\frac{z}{2}}}}{2 \cdot \sqrt[6]{64^{- 3}}} = 2 \cdot \sqrt[6]{64^{- 3}} \cdot 4$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $2 \cdot \sqrt[6]{64^{- 3}}$ .

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Step 2.1.1.1

Cancel the common factor.

$2 \cdot \sqrt[6]{64^{- 3}} \frac{\sqrt[3]{16^{\frac{z}{2}}}}{2 \cdot \sqrt[6]{64^{- 3}}} = 2 \cdot \sqrt[6]{64^{- 3}} \cdot 4$

Step 2.1.1.2

Rewrite the expression.

$\sqrt[3]{16^{\frac{z}{2}}} = 2 \cdot \sqrt[6]{64^{- 3}} \cdot 4$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $2 \cdot \sqrt[6]{64^{- 3}} \cdot 4$ .

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Step 2.2.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\sqrt[3]{16^{\frac{z}{2}}} = 2 \cdot \sqrt[6]{\frac{1}{64^{3}}} \cdot 4$

Step 2.2.1.2

Raise $64$ to the power of $3$ .

$\sqrt[3]{16^{\frac{z}{2}}} = 2 \cdot \sqrt[6]{\frac{1}{262144}} \cdot 4$

Step 2.2.1.3

Rewrite $\sqrt[6]{\frac{1}{262144}}$ as $\frac{\sqrt[6]{1}}{\sqrt[6]{262144}}$ .

$\sqrt[3]{16^{\frac{z}{2}}} = 2 \cdot \frac{\sqrt[6]{1}}{\sqrt[6]{262144}} \cdot 4$

Step 2.2.1.4

Any root of $1$ is $1$ .

$\sqrt[3]{16^{\frac{z}{2}}} = 2 \cdot \frac{1}{\sqrt[6]{262144}} \cdot 4$

Step 2.2.1.5

Simplify the denominator.

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Step 2.2.1.5.1

Rewrite $262144$ as $8^{6}$ .

$\sqrt[3]{16^{\frac{z}{2}}} = 2 \cdot \frac{1}{\sqrt[6]{8^{6}}} \cdot 4$

Step 2.2.1.5.2

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

$\sqrt[3]{16^{\frac{z}{2}}} = 2 \cdot \frac{1}{8} \cdot 4$

Step 2.2.1.6

Reduce the expression by cancelling the common factors.

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Step 2.2.1.6.1

Cancel the common factor of $2$ .

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Step 2.2.1.6.1.1

Factor $2$ out of $8$ .

$\sqrt[3]{16^{\frac{z}{2}}} = 2 \cdot \frac{1}{2 (4)} \cdot 4$

Step 2.2.1.6.1.2

Cancel the common factor.

$\sqrt[3]{16^{\frac{z}{2}}} = 2 \cdot \frac{1}{2 \cdot 4} \cdot 4$

Step 2.2.1.6.1.3

Rewrite the expression.

$\sqrt[3]{16^{\frac{z}{2}}} = \frac{1}{4} \cdot 4$

Step 2.2.1.6.2

Cancel the common factor of $4$ .

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Step 2.2.1.6.2.1

Cancel the common factor.

$\sqrt[3]{16^{\frac{z}{2}}} = \frac{1}{4} \cdot 4$

Step 2.2.1.6.2.2

Rewrite the expression.

$\sqrt[3]{16^{\frac{z}{2}}} = 1$

Step 3

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

${\sqrt[3]{16^{\frac{z}{2}}}}^{3} = 1^{3}$

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 $\sqrt[3]{16^{\frac{z}{2}}}$ as $16^{\frac{\frac{z}{2}}{3}}$ .

${(16^{\frac{\frac{z}{2}}{3}})}^{3} = 1^{3}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

${(16^{\frac{z}{2} \cdot \frac{1}{3}})}^{3} = 1^{3}$

Step 4.3

Multiply $\frac{z}{2} \cdot \frac{1}{3}$ .

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Step 4.3.1

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

${(16^{\frac{z}{2 \cdot 3}})}^{3} = 1^{3}$

Step 4.3.2

Multiply $2$ by $3$ .

${(16^{\frac{z}{6}})}^{3} = 1^{3}$

Step 4.4

Simplify the left side.

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Step 4.4.1

Multiply the exponents in ${(16^{\frac{z}{6}})}^{3}$ .

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Step 4.4.1.1

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

$16^{\frac{z}{6} \cdot 3} = 1^{3}$

Step 4.4.1.2

Cancel the common factor of $3$ .

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Step 4.4.1.2.1

Factor $3$ out of $6$ .

$16^{\frac{z}{3 (2)} \cdot 3} = 1^{3}$

Step 4.4.1.2.2

Cancel the common factor.

$16^{\frac{z}{3 \cdot 2} \cdot 3} = 1^{3}$

Step 4.4.1.2.3

Rewrite the expression.

$16^{\frac{z}{2}} = 1^{3}$

Step 4.5

Simplify the right side.

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Step 4.5.1

One to any power is one.

$16^{\frac{z}{2}} = 1$

Step 5

Solve for $z$ .

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Step 5.1

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

$\ln (16^{\frac{z}{2}}) = \ln (1)$

Step 5.2

Expand the left side.

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Step 5.2.1

Expand $\ln (16^{\frac{z}{2}})$ by moving $\frac{z}{2}$ outside the logarithm.

$\frac{z}{2} \ln (16) = \ln (1)$

Step 5.2.2

Combine $\frac{z}{2}$ and $\ln (16)$ .

$\frac{z \ln (16)}{2} = \ln (1)$

Step 5.3

Simplify the right side.

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Step 5.3.1

The natural logarithm of $1$ is $0$ .

$\frac{z \ln (16)}{2} = 0$

Step 5.4

Set the numerator equal to zero.

$z \ln (16) = 0$

Step 5.5

Divide each term in $z \ln (16) = 0$ by $\ln (16)$ and simplify.

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Step 5.5.1

Divide each term in $z \ln (16) = 0$ by $\ln (16)$ .

$\frac{z \ln (16)}{\ln (16)} = \frac{0}{\ln (16)}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $\ln (16)$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{z \ln (16)}{\ln (16)} = \frac{0}{\ln (16)}$

Step 5.5.2.1.2

Divide $z$ by $1$ .

$z = \frac{0}{\ln (16)}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Rewrite $\ln (16)$ as $\ln (2^{4})$ .

$z = \frac{0}{\ln (2^{4})}$

Step 5.5.3.2

Expand $\ln (2^{4})$ by moving $4$ outside the logarithm.

$z = \frac{0}{4 \ln (2)}$

Step 5.5.3.3

Cancel the common factor of $0$ and $4$ .

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Step 5.5.3.3.1

Factor $4$ out of $0$ .

$z = \frac{4 (0)}{4 \ln (2)}$

Step 5.5.3.3.2

Cancel the common factors.

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Step 5.5.3.3.2.1

Factor $4$ out of $4 \ln (2)$ .

$z = \frac{4 (0)}{4 (\ln (2))}$

Step 5.5.3.3.2.2

Cancel the common factor.

$z = \frac{4 \cdot 0}{4 \ln (2)}$

Step 5.5.3.3.2.3

Rewrite the expression.

$z = \frac{0}{\ln (2)}$

Step 5.5.3.4

Divide $0$ by $\ln (2)$ .

$z = 0$