Algebra Examples

$\sqrt[3]{\frac{{(1 \times 10^{- 3})}^{2} (10^{4} \times 10^{2})}{{(10^{6})}^{- 1} {(10)}^{3}}}$

Step 1

Multiply $10^{4}$ by $10^{2}$ by adding the exponents.

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Step 1.1

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

$\sqrt[3]{\frac{{(1 \cdot 10^{- 3})}^{2} \cdot 10^{4 + 2}}{{(10^{6})}^{- 1} {(10)}^{3}}}$

Step 1.2

Add $4$ and $2$ .

$\sqrt[3]{\frac{{(1 \cdot 10^{- 3})}^{2} \cdot 10^{6}}{{(10^{6})}^{- 1} {(10)}^{3}}}$

Step 2

Multiply $10^{- 3}$ by $1$ .

$\sqrt[3]{\frac{{(10^{- 3})}^{2} \cdot 10^{6}}{{(10^{6})}^{- 1} {(10)}^{3}}}$

Step 3

Reduce the expression $\frac{{(10^{- 3})}^{2} \cdot 10^{6}}{{(10^{6})}^{- 1} {(10)}^{3}}$ by cancelling the common factors.

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Step 3.1

Factor $10^{3}$ out of ${(10^{- 3})}^{2} \cdot 10^{6}$ .

$\sqrt[3]{\frac{10^{3} ({(10^{- 3})}^{2} \cdot 10^{3})}{{(10^{6})}^{- 1} {(10)}^{3}}}$

Step 3.2

Factor $10^{3}$ out of ${(10^{6})}^{- 1} {(10)}^{3}$ .

$\sqrt[3]{\frac{10^{3} ({(10^{- 3})}^{2} \cdot 10^{3})}{{(10)}^{3} {(10^{6})}^{- 1}}}$

Step 3.3

Cancel the common factor.

$\sqrt[3]{\frac{10^{3} ({(10^{- 3})}^{2} \cdot 10^{3})}{{(10)}^{3} {(10^{6})}^{- 1}}}$

Step 3.4

Rewrite the expression.

$\sqrt[3]{\frac{{(10^{- 3})}^{2} \cdot 10^{3}}{{(10^{6})}^{- 1}}}$

Step 4

Move ${(10^{6})}^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\sqrt[3]{{(10^{- 3})}^{2} \cdot 10^{3} \cdot 10^{6}}$

Step 5

Multiply $10^{3}$ by $10^{6}$ by adding the exponents.

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Step 5.1

Move $10^{6}$ .

$\sqrt[3]{{(10^{- 3})}^{2} \cdot (10^{6} \cdot 10^{3})}$

Step 5.2

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

$\sqrt[3]{{(10^{- 3})}^{2} \cdot 10^{6 + 3}}$

Step 5.3

Add $6$ and $3$ .

$\sqrt[3]{{(10^{- 3})}^{2} \cdot 10^{9}}$

Step 6

Multiply the exponents in ${(10^{- 3})}^{2}$ .

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Step 6.1

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

$\sqrt[3]{10^{- 3 \cdot 2} \cdot 10^{9}}$

Step 6.2

Multiply $- 3$ by $2$ .

$\sqrt[3]{10^{- 6} \cdot 10^{9}}$

Step 7

Multiply $10^{- 6}$ by $10^{9}$ by adding the exponents.

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Step 7.1

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

$\sqrt[3]{10^{- 6 + 9}}$

Step 7.2

Add $- 6$ and $9$ .

$\sqrt[3]{10^{3}}$

Step 8

Raise $10$ to the power of $3$ .

$\sqrt[3]{1000}$

Step 9

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

$\sqrt[3]{10^{3}}$

Step 10

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

$10$