Calculus Examples

$f (x) = x^{0.666666}$

Step 1

Set $x^{0.666666}$ equal to $0$ .

$x^{0.666666} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Convert the decimal exponent to a fractional exponent.

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Step 2.1.1

Convert the decimal number to a fraction by placing the decimal number over a power of ten. Since there are $6$ numbers to the right of the decimal point, place the decimal number over $10^{6}$ $(1000000)$ . Next, add the whole number to the left of the decimal.

$x^{0 \frac{666666}{1000000}} = 0$

Step 2.1.2

Reduce the fraction.

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Step 2.1.2.1

Convert $0 \frac{666666}{1000000}$ to an improper fraction.

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Step 2.1.2.1.1

A mixed number is an addition of its whole and fractional parts.

$x^{0 + \frac{666666}{1000000}} = 0$

Step 2.1.2.1.2

Add $0$ and $\frac{666666}{1000000}$ .

$x^{\frac{666666}{1000000}} = 0$

Step 2.1.2.2

Cancel the common factor of $666666$ and $1000000$ .

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Step 2.1.2.2.1

Factor $2$ out of $666666$ .

$x^{\frac{2 (333333)}{1000000}} = 0$

Step 2.1.2.2.2

Cancel the common factors.

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Step 2.1.2.2.2.1

Factor $2$ out of $1000000$ .

$x^{\frac{2 \cdot 333333}{2 \cdot 500000}} = 0$

Step 2.1.2.2.2.2

Cancel the common factor.

$x^{\frac{2 \cdot 333333}{2 \cdot 500000}} = 0$

Step 2.1.2.2.2.3

Rewrite the expression.

$x^{\frac{333333}{500000}} = 0$

Step 2.2

Raise each side of the equation to the power of $\frac{1}{0.666666}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{333333}{500000}})}^{\frac{1}{0.666666}} = 0^{\frac{1}{0.666666}}$

Step 2.3

Simplify the exponent.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Simplify ${(x^{\frac{333333}{500000}})}^{\frac{1}{0.666666}}$ .

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Step 2.3.1.1.1

Multiply the exponents in ${(x^{\frac{333333}{500000}})}^{\frac{1}{0.666666}}$ .

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Step 2.3.1.1.1.1

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

$x^{\frac{333333}{500000} \cdot \frac{1}{0.666666}} = 0^{\frac{1}{0.666666}}$

Step 2.3.1.1.1.2

Cancel the common factor of $0.666666$ .

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Step 2.3.1.1.1.2.1

Factor $0.666666$ out of $333333$ .

$x^{\frac{0.666666 (500000)}{500000} \cdot \frac{1}{0.666666}} = 0^{\frac{1}{0.666666}}$

Step 2.3.1.1.1.2.2

Cancel the common factor.

$x^{\frac{0.666666 \cdot 500000}{500000} \cdot \frac{1}{0.666666}} = 0^{\frac{1}{0.666666}}$

Step 2.3.1.1.1.2.3

Rewrite the expression.

$x^{\frac{500000}{500000}} = 0^{\frac{1}{0.666666}}$

Step 2.3.1.1.1.3

Divide $500000$ by $500000$ .

$x^{1} = 0^{\frac{1}{0.666666}}$

Step 2.3.1.1.2

Simplify.

$x = 0^{\frac{1}{0.666666}}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $0^{\frac{1}{0.666666}}$ .

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Step 2.3.2.1.1

Divide $1$ by $0.666666$ .

$x = 0^{1.\overline{500001}}$

Step 2.3.2.1.2

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

$x = 0$

Step 3