PRACTICE WITH EXPONENTS
DEFINITIONS:   properties of exponents
base;
exponent;
power
Let [beautiful math coming... please be patient] $\,x\in\Bbb{R}\,$.
In the expression [beautiful math coming... please be patient] $\,x^n\,$,
$\,x\,$ is called the base
and $\,n\,$ is called the exponent or the power.
positive integers If [beautiful math coming... please be patient] $\,n\in\{1,2,3,\ldots\}\,$,
then [beautiful math coming... please be patient] $\,x^n = x\cdot x\cdot x \cdot \ldots \cdot x\,$,
where there are $\,n\,$ factors in the product.

In this case, [beautiful math coming... please be patient] $\,x^n\,$ is just a shorthand for repeated multiplication.

Note that [beautiful math coming... please be patient] $\,x^1 = x\,$ for all real numbers $\,x\,$.
zero If [beautiful math coming... please be patient] $\,x\ne 0\,$, then $\,x^0 = 1\,$.
The expression $\,0^0\,$ is not defined.
negative integers If [beautiful math coming... please be patient] $\,n\in\{1,2,3,\ldots\}\,$ and $\,x\ne 0\,$,
then [beautiful math coming... please be patient] $\displaystyle\,x^{-n} = \frac{1}{x^n} = \frac{1}{x\cdot x\cdot x\cdot \ldots \cdot x}$,
where there are $\,n\,$ factors in the product.
In particular, [beautiful math coming... please be patient] $\,\displaystyle x^{-1} = \frac{1}{x}\,$ for all nonzero real numbers $\,x\,$.
That is, [beautiful math coming... please be patient] $\,x^{-1}\,$ 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 [beautiful math coming... please be patient] $\,(-2)^6\,$.
There are an even number ($6$) of negative factors, so the result is positive.
The size of the result is $\,2^6 = 64\,$.
Thus, [beautiful math coming... please be patient] $\,(-2)^6 = 64\,$.

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

Since exponents are done before multiplication,
[beautiful math coming... please be patient] $\,-2^4 = (-1)(2^4) = (-1)(16) = -16\,$.

Be careful!

The numbers [beautiful math coming... please be patient] $\,-2^4\,$ and $\,(-2)^4\,$ represent different orders of operations,
and are different numbers!

The numbers [beautiful math coming... please be patient] $\,-2^3\,$ and $\,(-2)^3\,$ represent different orders of operations,
but in this case they result in the same number!

EXAMPLES:
[beautiful math coming... please be patient] $(-2)^3 = -8$
[beautiful math coming... please be patient] $-2^3 = -8$
[beautiful math coming... please be patient] $(-2)^4 = 16$
[beautiful math coming... please be patient] $-2^4 = -16$
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Practice with Order of Operations

 
 

If an expression is not defined, input “nd”.

Simplify:
    
(an even number, please)