RATIONAL EXPONENTS

Algebra One

Explore in your Notebook…• Evaluate the following:a) b) c) – d)

e) f) g) h)

• What is an irrational number? • What is a rational number?• How can we predict/determine when a

square root is irrational or rational?• List the first 15 perfect squares.

121 15

04.025.6

281

25

9784

Perfect Squares

It’s important to know your perfect squares – they can be useful for estimating values for irrational square roots.

Estimate the following square roots without a calculator:

14 35 110

IMPORTANT FACT

SQUARING and SQUARE-ROOTING are INVERSE OPERATIONS.

Can we take roots other than “square roots”?

Yes we can take any kind of root – for example:

The cube root of 8 is noted as

The fifth root of 32 is noted as

3 8

5 32

SQUARE ROOTS & Nth ROOTS

Yes we can take any kind of root – for example:

The square root of any number:

The nth root of any number:

*the “index” of the radical tells you what root you are taking, if you don’t see an “n” then it is square root.

aa =2

aan n =

What does this have to do with EXPONENTS?

Consider… roots and powers are inverse operations

square root squaring

Cube root cubing

Fourth root fourth power

Nth root nth power

aa =2

aa =3 3

aa =4 4

aan n =

↔↔

↔

↔

If you could turn a “root” into a power, what would it look like?

Remember – inverse operations “cancel” out.

so is the same as

so is the same as

so is the same as

so is the same as

aa =2

aa =3 3

aa =4 4

aan n =

2a

3 3a

4 4a

n na

RATIONAL EXPONENTS

The nth root of a positive number can be written as a power with base “a” and exponent “1/n”

nn aa /1=

RATIONAL EXPONENTS

This makes nth roots very easy to evaluate on our calculator, just remember to put parentheses around the full exponent.

4/14 81=81 3/13 216=216

2/1289=289

Get comfortable going back & forth between radical & exponential notation for nth roots.

• Write the following using rational exponent notation:

a) b) • Write the following using radical notation.c) d) 61/9 e) Seventh root(s) of 13

What about Rational Exponents that do NOT have a numerator of ONE?

• What does this mean? 163/4

consider reversing the power of a power property 163/4 = 1631/4 = (163)1/4

So what does that numerator represent?

RATIONAL EXPONENTS

The nth of a positive number can be written as a power with base “a” and exponent “1/n”

( ) nmm

nn m aaa /==

This makes it quite easy to evaluate on your calculator if you remember how to

rewrite them!

( ) =1634 ( ) =27

43

( ) =95 ( ) =256

34

Cool Down – THINK ABOUT IT…

What is the meaning of a negative rational exponent?

8-4/3

Homework

Worksheet – Radicals & Rational Exponents