# Practice with Exponents

## DEFINITIONS: Properties of Exponents

*base*and $\,n\,$ is called the

*exponent*or the

*power*.

If $\,n\in\{1,2,3,\ldots\}\,,$ then $$\cssId{s15}{x^n = x\cdot x\cdot x \cdot \ldots \cdot x\,,}$$ where there are $\,n\,$ factors in the product.

In this case, $\,x^n\,$ is just a shorthand for repeated multiplication.

Note that $\,x^1 = x\,$ for all real numbers $\,x\,.$

If $\,x\ne 0\,,$ then $\,x^0 = 1\,.$

The expression $\,0^0\,$ is not defined.

If $\,n\in\{1,2,3,\ldots\}\,$ and $\,x\ne 0\,,$ then $$\cssId{s25}{x^{-n} = \frac{1}{x^n} = \frac{1}{x\cdot x\cdot x\cdot \ldots \cdot x}\,,}$$ where there are $\,n\,$ factors in the product.

In particular, $\,\displaystyle x^{-1} = \frac{1}{x}\,$ for all nonzero real numbers $\,x\,.$ That is, $\,x^{-1}\,$ is the reciprocal of $\,x\,.$

When simplifying expressions involving exponent notation,
figure out the *sign* (plus or minus) of the expression first,
then figure out its *size*.

Recall that any even number ($2$, $4$, $6$, $\ldots$) of negative factors is positive. Any odd number ($1$, $3$, $5$, $\ldots$) of negative factors is negative.

For example, consider $\,(-2)^6\,.$ There are an even number ($6$) of negative factors, so the result is positive. The size of the result is $\,2^6 = 64\,.$ Thus, $\,(-2)^6 = 64\,.$

As a second example, consider $\,(-2)^5\,.$ There are an odd number ($5$) of negative factors, so the result is negative. The size of the result is $\,2^5 = 32\,.$ Thus, $\,(-2)^5 = -32\,.$

Since exponents are done before multiplication: $$ \cssId{s49}{-2^4} \cssId{s50}{= (-1)(2^4)} \cssId{s51}{= (-1)(16)} \cssId{s52}{= -16} $$

Be careful!

The numbers $\,-2^4\,$ and $\,(-2)^4\,$ represent different orders of operations, and are different numbers!

The numbers $\,-2^3\,$ and $\,(-2)^3\,$ represent different orders of operations, but in this case they result in the same number!

## Examples

## Practice

If an expression is not defined, input ‘nd’.