Basic Math Examples

$10^{- 3 w} \cdot 10^{w} = \frac{1}{10}$

Step 1

Raise $10$ to the power of $1$ .

$10^{- 3 w} \cdot 10^{w} = \frac{1}{10^{1}}$

Step 2

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

$10^{- 3 w} \cdot 10^{w} = 10^{- 1}$

Step 3

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

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Step 3.1

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

$10^{- 3 w + w} = 10^{- 1}$

Step 3.2

Add $- 3 w$ and $w$ .

$10^{- 2 w} = 10^{- 1}$

Step 4

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$- 2 w = - 1$

Step 5

Divide each term in $- 2 w = - 1$ by $- 2$ and simplify.

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Step 5.1

Divide each term in $- 2 w = - 1$ by $- 2$ .

$\frac{- 2 w}{- 2} = \frac{- 1}{- 2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 2 w}{- 2} = \frac{- 1}{- 2}$

Step 5.2.1.2

Divide $w$ by $1$ .

$w = \frac{- 1}{- 2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Dividing two negative values results in a positive value.

$w = \frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$w = \frac{1}{2}$

Decimal Form:

$w = 0.5$