﻿ Practice with Exponents
PRACTICE WITH EXPONENTS
by Dr. Carol JVF Burns (website creator)
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• PRACTICE (online exercises and printable worksheets)
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 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 $\,x^n = x\cdot x\cdot x \cdot \ldots \cdot x\,$, 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 $\displaystyle\,x^{-n} = \frac{1}{x^n} = \frac{1}{x\cdot x\cdot x\cdot \ldots \cdot x}$, where there are $\,n\,$ factors in the product. In particular, $\,\displaystyle x^{-1} = \frac{1}{x}\,$ 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:
$(-2)^3 = -8$
$-2^3 = -8$
$(-2)^4 = 16$
$-2^4 = -16$
Master the ideas from this section