Consider the following chart that shows the expansion of $a$ for several exponents:

$\begin{array}{lllllllll}{a}^4\phantom{-}& =& a& \times & a& \times & a& \times & a\\ a^3& =& a& \times & a& \times & a& & \\ a^2& =& a& \times & a& & & & \\ a^1& =& a& & & & & & \\ a^0& =& 1& & & & & & \end{array}$

$\begin{array}{lll}{a}^{-1}& =& {\displaystyle \frac{1}{a}}\\ \\ a^{-2}& =& {\displaystyle \frac{1}{(a\times a)}}\\ \\ a^{-3}& =& {\displaystyle \frac{1}{(a\times a\times a)}}\\ \\ a^{-4}& =& {\displaystyle \frac{1}{(a\times a\times a\times a)}}\end{array}$

If zero and negative exponents are expanded to base 2, the result is the following:

$\begin{array}{llcll}{2}^0& =& 1& & \\ \\ 2^{-1}& =& {\displaystyle \frac{1}{2}}& & \\ \\ 2^{-2}& =& {\displaystyle \frac{1}{2\times 2}}& \text{or}& {\displaystyle \frac{1}{4}}\\ \\ 2^{-3}& =& {\displaystyle \frac{1}{2\times 2\times 2}}& \text{or}& {\displaystyle \frac{1}{8}}\\ \\ 2^{-4}& =& {\displaystyle \frac{1}{2\times 2\times 2\times 2}}& \text{or}& {\displaystyle \frac{1}{16}}\end{array}$

The most unusual of these is the exponent 0. Any base that is not equal to zero to the zeroth exponent is always 1. The simplest explanation of this is by example.

This zero rule of exponents can make difficult problems elementary simply because whatever the 0 exponent is attached to reduces to 1. Consider the following examples:

When encountering these types of problems, always remain aware of what the zero power is attached to, since only what it is attached to cancels to 1.

When dealing with negative exponents, the simplest solution is to reciprocate the power. For instance:

Four Rules of Negative Exponents

$x^{-n}={\displaystyle \frac{1}{x^n}}(x\ne 0)$

${\displaystyle \frac{1}{x^{-n}}}=x^n(x\ne 0)$

${\left({\displaystyle \frac{x}{y}}\right)}^{-n}={\left({\displaystyle \frac{y}{x}}\right)}^n(x,y\ne 0)$

${\left({\displaystyle \frac{x^a{y}^b}{z^c}}\right)}^{-n}={\displaystyle \frac{z^{cn}}{x^{an}{y}^{bn}}}(x,y,z\ne 0)$

Questions

Simplify. Your answer should contain only positive exponents.

1. $(2x^4{y}^{-2})(2x{y}^3{)}^4$
2. $(2a^{-2}{b}^{-3})(2a^0{b}^4{)}^4$
3. $(2x^2{y}^2{)}^4{x}^{-4}$
4. $[(m^0{n}^3)(2m^{-3}{n}^{-3}){]}^0$
5. $(2x^{-3}{y}^2)\mathbf{\nabla }\mathbf{\cdot }(3x^{-3}{y}^3\cdot 3x^0)$
6. $3y^3\mathbf{\nabla }\mathbf{\cdot }[(3y{x}^3)(2x^4{y}^{-3})]$
7. $2y\mathbf{\nabla }\mathbf{\cdot }{\left(x^0{y}^2\right)}^4$
8. $(a^4{)}^4\mathbf{\nabla }\mathbf{\cdot }2b$
9. $(2a^2{b}^3{)}^4\mathbf{\nabla }\mathbf{\cdot }{a}^{-1}$
10. $(2y^{-4}{)}^{-2}\mathbf{\nabla }\mathbf{\cdot }{x}^2$
11. $(2mn{)}^4\mathbf{\nabla }\mathbf{\cdot }{m}^0{n}^{-2}$
12. $2x^{-3}\mathbf{\nabla }\mathbf{\cdot }{\left(x^4{y}^{-3}\right)}^{-1}$
13. $[(2u^{-2}{v}^3)(2u{v}^4{)}^{-1}]\mathbf{\nabla }\mathbf{\cdot }2u^{-4}{v}^0$
14. $[(2y{x}^2)(x^{-2})]\mathbf{\nabla }\mathbf{\cdot }{\left(2x^0{y}^4\right)}^{-1}$
15. $b^{-1}\mathbf{\nabla }\mathbf{\cdot }[(2a^4{b}^0{)}^0(2a^{-3}{b}^2)]$
16. $2yz{x}^2\mathbf{\nabla }\mathbf{\cdot }[(2x^4{y}^4{z}^{-2})(z{y}^2{)}^4]$
17. $[(c{b}^3{)}^2(2a^{-3}{b}^2)]\mathbf{\nabla }\mathbf{\cdot }{\left(a^3{b}^{-2}{c}^3\right)}^3$
18. $2q^4(m^2{p}^{2}q)\mathbf{\nabla }\mathbf{\cdot }{(2m\cdot 4p^2)}^3$
19. $(y{x}^{-4}{z}^2{)}^{-1}\mathbf{\nabla }\mathbf{\cdot }{z}^3\cdot x^2{y}^3{z}^{-1}$
20. $2mp{n}^{-3}\mathbf{\nabla }\mathbf{\cdot }[2n^2{p}^0(m^0{n}^{-4}{p}^2{)}^3]$

Answer Key 6.2