Calculus Examples

$16 x + 2 x^{- \frac{1}{2}}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $16 x + 2 x^{- \frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [16 x] + \frac{d}{d x} [2 x^{- \frac{1}{2}}]$ .

$\frac{d}{d x} [16 x] + \frac{d}{d x} [2 x^{- \frac{1}{2}}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [16 x]$ .

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Step 1.1.2.1

Since $16$ is constant with respect to $x$ , the derivative of $16 x$ with respect to $x$ is $16 \frac{d}{d x} [x]$ .

$16 \frac{d}{d x} [x] + \frac{d}{d x} [2 x^{- \frac{1}{2}}]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$16 \cdot 1 + \frac{d}{d x} [2 x^{- \frac{1}{2}}]$

Step 1.1.2.3

Multiply $16$ by $1$ .

$16 + \frac{d}{d x} [2 x^{- \frac{1}{2}}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [2 x^{- \frac{1}{2}}]$ .

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Step 1.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{- \frac{1}{2}}$ with respect to $x$ is $2 \frac{d}{d x} [x^{- \frac{1}{2}}]$ .

$16 + 2 \frac{d}{d x} [x^{- \frac{1}{2}}]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{1}{2}$ .

$16 + 2 (- \frac{1}{2} x^{- \frac{1}{2} - 1})$

Step 1.1.3.3

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

$16 + 2 (- \frac{1}{2} x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 1.1.3.4

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

$16 + 2 (- \frac{1}{2} x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 1.1.3.5

Combine the numerators over the common denominator.

$16 + 2 (- \frac{1}{2} x^{\frac{- 1 - 1 \cdot 2}{2}})$

Step 1.1.3.6

Simplify the numerator.

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Step 1.1.3.6.1

Multiply $- 1$ by $2$ .

$16 + 2 (- \frac{1}{2} x^{\frac{- 1 - 2}{2}})$

Step 1.1.3.6.2

Subtract $2$ from $- 1$ .

$16 + 2 (- \frac{1}{2} x^{\frac{- 3}{2}})$

Step 1.1.3.7

Move the negative in front of the fraction.

$16 + 2 (- \frac{1}{2} x^{- \frac{3}{2}})$

Step 1.1.3.8

Combine $x^{- \frac{3}{2}}$ and $\frac{1}{2}$ .

$16 + 2 (- \frac{x^{- \frac{3}{2}}}{2})$

Step 1.1.3.9

Multiply $- 1$ by $2$ .

$16 - 2 \frac{x^{- \frac{3}{2}}}{2}$

Step 1.1.3.10

Combine $- 2$ and $\frac{x^{- \frac{3}{2}}}{2}$ .

$16 + \frac{- 2 x^{- \frac{3}{2}}}{2}$

Step 1.1.3.11

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$16 + \frac{- 2}{2 x^{\frac{3}{2}}}$

Step 1.1.3.12

Factor $2$ out of $- 2$ .

$16 + \frac{2 \cdot - 1}{2 x^{\frac{3}{2}}}$

Step 1.1.3.13

Cancel the common factors.

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Step 1.1.3.13.1

Factor $2$ out of $2 x^{\frac{3}{2}}$ .

$16 + \frac{2 \cdot - 1}{2 (x^{\frac{3}{2}})}$

Step 1.1.3.13.2

Cancel the common factor.

$16 + \frac{2 \cdot - 1}{2 x^{\frac{3}{2}}}$

Step 1.1.3.13.3

Rewrite the expression.

$16 + \frac{- 1}{x^{\frac{3}{2}}}$

Step 1.1.3.14

Move the negative in front of the fraction.

$f' (x) = 16 - \frac{1}{x^{\frac{3}{2}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $16 - \frac{1}{x^{\frac{3}{2}}}$ .

$16 - \frac{1}{x^{\frac{3}{2}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $16 - \frac{1}{x^{\frac{3}{2}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$16 - \frac{1}{x^{\frac{3}{2}}} = 0$

Step 2.2

Subtract $16$ from both sides of the equation.

$- \frac{1}{x^{\frac{3}{2}}} = - 16$

Step 2.3

Find the LCD of the terms in the equation.

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Step 2.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{\frac{3}{2}}, 1$

Step 2.3.2

The LCM of one and any expression is the expression.

$x^{\frac{3}{2}}$

Step 2.4

Multiply each term in $- \frac{1}{x^{\frac{3}{2}}} = - 16$ by $x^{\frac{3}{2}}$ to eliminate the fractions.

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Step 2.4.1

Multiply each term in $- \frac{1}{x^{\frac{3}{2}}} = - 16$ by $x^{\frac{3}{2}}$ .

$- \frac{1}{x^{\frac{3}{2}}} x^{\frac{3}{2}} = - 16 x^{\frac{3}{2}}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $x^{\frac{3}{2}}$ .

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Step 2.4.2.1.1

Move the leading negative in $- \frac{1}{x^{\frac{3}{2}}}$ into the numerator.

$\frac{- 1}{x^{\frac{3}{2}}} x^{\frac{3}{2}} = - 16 x^{\frac{3}{2}}$

Step 2.4.2.1.2

Cancel the common factor.

$\frac{- 1}{x^{\frac{3}{2}}} x^{\frac{3}{2}} = - 16 x^{\frac{3}{2}}$

Step 2.4.2.1.3

Rewrite the expression.

$- 1 = - 16 x^{\frac{3}{2}}$

Step 2.5

Solve the equation.

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Step 2.5.1

Rewrite the equation as $- 16 x^{\frac{3}{2}} = - 1$ .

$- 16 x^{\frac{3}{2}} = - 1$

Step 2.5.2

Divide each term in $- 16 x^{\frac{3}{2}} = - 1$ by $- 16$ and simplify.

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Step 2.5.2.1

Divide each term in $- 16 x^{\frac{3}{2}} = - 1$ by $- 16$ .

$\frac{- 16 x^{\frac{3}{2}}}{- 16} = \frac{- 1}{- 16}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor.

$\frac{- 16 x^{\frac{3}{2}}}{- 16} = \frac{- 1}{- 16}$

Step 2.5.2.2.2

Divide $x^{\frac{3}{2}}$ by $1$ .

$x^{\frac{3}{2}} = \frac{- 1}{- 16}$

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Dividing two negative values results in a positive value.

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

Step 2.5.3

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

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

Step 2.5.4

Simplify the exponent.

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Step 2.5.4.1

Simplify the left side.

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Step 2.5.4.1.1

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

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Step 2.5.4.1.1.1

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

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Step 2.5.4.1.1.1.1

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

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

Step 2.5.4.1.1.1.2

Cancel the common factor of $3$ .

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Step 2.5.4.1.1.1.2.1

Cancel the common factor.

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

Step 2.5.4.1.1.1.2.2

Rewrite the expression.

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

Step 2.5.4.1.1.1.3

Cancel the common factor of $2$ .

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Step 2.5.4.1.1.1.3.1

Cancel the common factor.

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

Step 2.5.4.1.1.1.3.2

Rewrite the expression.

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

Step 2.5.4.1.1.2

Simplify.

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

Step 2.5.4.2

Simplify the right side.

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Step 2.5.4.2.1

Simplify ${(\frac{1}{16})}^{\frac{2}{3}}$ .

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Step 2.5.4.2.1.1

Apply the product rule to $\frac{1}{16}$ .

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

Step 2.5.4.2.1.2

One to any power is one.

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

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$16 - \frac{1}{\sqrt{x^{3}}}$

Step 3.2

Set the denominator in $\frac{1}{\sqrt{x^{3}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{x^{3}} = 0$

Step 3.3

Solve for $x$ .

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Step 3.3.1

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

${\sqrt{x^{3}}}^{2} = 0^{2}$

Step 3.3.2

Simplify each side of the equation.

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Step 3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3}}$ as $x^{\frac{3}{2}}$ .

${(x^{\frac{3}{2}})}^{2} = 0^{2}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

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

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Step 3.3.2.2.1.1

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

$x^{\frac{3}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.2.1.2.1

Cancel the common factor.

$x^{\frac{3}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.2.2

Rewrite the expression.

$x^{3} = 0^{2}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$x^{3} = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

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

$x = \sqrt[3]{0}$

Step 3.3.3.2

Simplify $\sqrt[3]{0}$ .

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Step 3.3.3.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 3.3.3.2.2

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

$x = 0$

Step 3.4

Set the radicand in $\sqrt{x^{3}}$ less than $0$ to find where the expression is undefined.

$x^{3} < 0$

Step 3.5

Solve for $x$ .

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Step 3.5.1

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

$\sqrt[3]{x^{3}} < \sqrt[3]{0}$

Step 3.5.2

Simplify the equation.

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Step 3.5.2.1

Simplify the left side.

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Step 3.5.2.1.1

Pull terms out from under the radical.

$x < \sqrt[3]{0}$

Step 3.5.2.2

Simplify the right side.

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Step 3.5.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 3.5.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$x < \sqrt[3]{0^{3}}$

Step 3.5.2.2.1.2

Pull terms out from under the radical.

$x < 0$

Step 3.6

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq 0$

$(- \infty, 0]$

$x \leq 0$

$(- \infty, 0]$

Step 4

Evaluate $16 x + 2 x^{- \frac{1}{2}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{1}{16^{\frac{2}{3}}}$ .

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Step 4.1.1

Substitute $\frac{1}{16^{\frac{2}{3}}}$ for $x$ .

$16 (\frac{1}{16^{\frac{2}{3}}}) + 2 {(\frac{1}{16^{\frac{2}{3}}})}^{- \frac{1}{2}}$

Step 4.1.2

Simplify each term.

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Step 4.1.2.1

Combine $16$ and $\frac{1}{16^{\frac{2}{3}}}$ .

$\frac{16}{16^{\frac{2}{3}}} + 2 {(\frac{1}{16^{\frac{2}{3}}})}^{- \frac{1}{2}}$

Step 4.1.2.2

Move $16^{\frac{2}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$16 \cdot 16^{- \frac{2}{3}} + 2 {(\frac{1}{16^{\frac{2}{3}}})}^{- \frac{1}{2}}$

Step 4.1.2.3

Multiply $16$ by $16^{- \frac{2}{3}}$ by adding the exponents.

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Step 4.1.2.3.1

Multiply $16$ by $16^{- \frac{2}{3}}$ .

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Step 4.1.2.3.1.1

Raise $16$ to the power of $1$ .

$16^{1} \cdot 16^{- \frac{2}{3}} + 2 {(\frac{1}{16^{\frac{2}{3}}})}^{- \frac{1}{2}}$

Step 4.1.2.3.1.2

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

$16^{1 - \frac{2}{3}} + 2 {(\frac{1}{16^{\frac{2}{3}}})}^{- \frac{1}{2}}$

Step 4.1.2.3.2

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

$16^{\frac{3}{3} - \frac{2}{3}} + 2 {(\frac{1}{16^{\frac{2}{3}}})}^{- \frac{1}{2}}$

Step 4.1.2.3.3

Combine the numerators over the common denominator.

$16^{\frac{3 - 2}{3}} + 2 {(\frac{1}{16^{\frac{2}{3}}})}^{- \frac{1}{2}}$

Step 4.1.2.3.4

Subtract $2$ from $3$ .

$16^{\frac{1}{3}} + 2 {(\frac{1}{16^{\frac{2}{3}}})}^{- \frac{1}{2}}$

Step 4.1.2.4

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 4.1.2.5

Multiply the exponents in ${(16^{\frac{2}{3}})}^{\frac{1}{2}}$ .

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Step 4.1.2.5.1

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

$16^{\frac{1}{3}} + 2 \cdot 16^{\frac{2}{3} \cdot \frac{1}{2}}$

Step 4.1.2.5.2

Cancel the common factor of $2$ .

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Step 4.1.2.5.2.1

Cancel the common factor.

$16^{\frac{1}{3}} + 2 \cdot 16^{\frac{2}{3} \cdot \frac{1}{2}}$

Step 4.1.2.5.2.2

Rewrite the expression.

$16^{\frac{1}{3}} + 2 \cdot 16^{\frac{1}{3}}$

Step 4.1.2.6

Multiply $2 \cdot 16^{\frac{1}{3}}$ .

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Step 4.1.2.6.1

Rewrite $16$ as $2^{4}$ .

$16^{\frac{1}{3}} + 2 \cdot {(2^{4})}^{\frac{1}{3}}$

Step 4.1.2.6.2

Multiply the exponents in ${(2^{4})}^{\frac{1}{3}}$ .

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Step 4.1.2.6.2.1

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

$16^{\frac{1}{3}} + 2 \cdot 2^{4 (\frac{1}{3})}$

Step 4.1.2.6.2.2

Combine $4$ and $\frac{1}{3}$ .

$16^{\frac{1}{3}} + 2 \cdot 2^{\frac{4}{3}}$

Step 4.1.2.6.3

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

$16^{\frac{1}{3}} + 2^{1 + \frac{4}{3}}$

Step 4.1.2.6.4

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

$16^{\frac{1}{3}} + 2^{\frac{3}{3} + \frac{4}{3}}$

Step 4.1.2.6.5

Combine the numerators over the common denominator.

$16^{\frac{1}{3}} + 2^{\frac{3 + 4}{3}}$

Step 4.1.2.6.6

Add $3$ and $4$ .

$16^{\frac{1}{3}} + 2^{\frac{7}{3}}$

Step 4.2

Evaluate at $x = 0$ .

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Step 4.2.1

Substitute $0$ for $x$ .

$16 (0) + 2 {(0)}^{- \frac{1}{2}}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Multiply $16$ by $0$ .

$0 + 2 {(0)}^{- \frac{1}{2}}$

Step 4.2.2.1.2

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

$0 + 2 \frac{1}{0^{\frac{1}{2}}}$

Step 4.2.2.1.3

Simplify the denominator.

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Step 4.2.2.1.3.1

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

$0 + 2 \frac{1}{{(0^{2})}^{\frac{1}{2}}}$

Step 4.2.2.1.3.2

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

$0 + 2 \frac{1}{0^{2 (\frac{1}{2})}}$

Step 4.2.2.1.3.3

Cancel the common factor of $2$ .

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Step 4.2.2.1.3.3.1

Cancel the common factor.

$0 + 2 \frac{1}{0^{2 (\frac{1}{2})}}$

Step 4.2.2.1.3.3.2

Rewrite the expression.

$0 + 2 \frac{1}{0^{1}}$

Step 4.2.2.1.3.4

Evaluate the exponent.

$0 + 2 (\frac{1}{0})$

Step 4.2.2.1.3.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$0 + 2 (\frac{1}{0})$

Step 4.2.2.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$0 + 2 (\frac{1}{0})$

Step 4.2.2.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.3

List all of the points.

$(\frac{1}{16^{\frac{2}{3}}}, 16^{\frac{1}{3}} + 2^{\frac{7}{3}})$

Step 5