Calculus Examples

$\frac{1000^{\frac{3}{6}}}{1000^{\frac{1}{6}}}$

Step 1

Move $1000^{\frac{1}{6}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$1000^{\frac{3}{6}} \cdot 1000^{- \frac{1}{6}}$

Step 2

Multiply $1000^{\frac{3}{6}}$ by $1000^{- \frac{1}{6}}$ by adding the exponents.

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Step 2.1

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

$1000^{\frac{3}{6} - \frac{1}{6}}$

Step 2.2

Combine the numerators over the common denominator.

$1000^{\frac{3 - 1}{6}}$

Step 2.3

Subtract $1$ from $3$ .

$1000^{\frac{2}{6}}$

Step 2.4

Cancel the common factor of $2$ and $6$ .

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Step 2.4.1

Factor $2$ out of $2$ .

$1000^{\frac{2 (1)}{6}}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $2$ out of $6$ .

$1000^{\frac{2 \cdot 1}{2 \cdot 3}}$

Step 2.4.2.2

Cancel the common factor.

$1000^{\frac{2 \cdot 1}{2 \cdot 3}}$

Step 2.4.2.3

Rewrite the expression.

$1000^{\frac{1}{3}}$

Step 3

Simplify the expression.

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Step 3.1

Rewrite $1000$ as $10^{3}$ .

${(10^{3})}^{\frac{1}{3}}$

Step 3.2

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

$10^{3 (\frac{1}{3})}$

Step 4

Cancel the common factor of $3$ .

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Step 4.1

Cancel the common factor.

$10^{3 (\frac{1}{3})}$

Step 4.2

Rewrite the expression.

$10^{1}$

Step 5

Evaluate the exponent.

$10$