audio read-through Practice with Exponents

DEFINITIONS: Properties of Exponents

base; exponent; power
Let $\,x\in\Bbb{R}\,$. In the expression $\,x^n\,$, $\,x\,$ is called the base and $\,n\,$ is called the exponent or the power.
positive integers

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 $\,x\,$.

zero

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

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

negative integers

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

$(-2)^3 = -8$
$-2^3 = -8$
$(-2)^4 = 16$
$-2^4 = -16$

Practice

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