Exponents with Negatives and Fractions Peter Gibby I

Exponents with Negatives and Fractions Peter Gibby

I <3 Fractions and Negatives I <3 Exponents! Lets put them together • We all love fractions so much we decided that we can put them into exponents! • But wait it gets better: we can use them to get rid of those pesky radicals. I hate typing out that radical sign, and now I don’t have to! You guys just gotta get used to fractional exponents!

But Really… • Yes we use fractions as exponents. We use them to represent roots. • When we have something to the second power, it means I multiply something twice. A fraction, like ½ is the same as, I multiply this one half times. • What on Earth!? How do we multiply something half a time?

Confused yet? • I hope not anyways. I hope that this will all make sense soon. Lets look at multiplying exponents to better understand this. • 22 x 24=(2 x 2)x(2 x 2 x 2 x 2)=2 x 2 x 2 x 2=26 • 5^4 x 5^4=(5 x 5 x 5 x 5)x(5 x 5 x 5 x 5)=5 x 5 x 5 x 5 x 5=5^8 • Notice how when we multiply exponents, we just add the exponents together? 2^2 x 2^4=2^6 and 5^4 x 5^4=5^8

So What About Exponents with Exponents? • Lets look at some exponents within exponents: – (44)2=44 x 44=44+4=48 – (23)5=23 x 23 x 23=23+3+3=215 • Notice that for both, we end up multiplying the two exponents together. 4 x 2=8 and 3 x 5=15

Lets Look at Radicals and Exponents • √ 64 = √ 82 =8^1 • √ 16 = √ 42 = √ 24 =2^2 • 3√ 64 =3√ 43 =4^1 • Notice that we end up dividing our exponent by the power of our root… • So √ 64 = √ 82 =81=(82)1/2 • Or more simply put: √ 64 = 641/2 • This quite simply shows that a fractional exponent equals a root.

What About Division? • Division with exponents is a fairly simple idea, and it is where negative exponents come from. • (42)/(41)=(4 x 4)/(4)=4 • Hmmm Notice that just like multiplying adds onto the exponent, division subtracts them. • So what if I did this? – (41)/(42)=(4)/(4 x 4)=1/4=41 -2=4 -1

Negative Exponents • We use them to say the reciprocal, or number that multiplied by the base that gives us 1. – 1/4 x 4=1 so 4 is the reciprocal of ¼ and ¼ is the reciprocal of 4

Lets Go Farther • (5^2)/(5^6)=5^(2 -6)=5^(-4)=1/(5^4) • 82/48=(43)2/48=4(3 x 2)-8=4 -2=1/16 – You can leave it either as a negative exponent, as 1 over a positive exponent or 1 over a number, as long as they are all equal.