Radical Operations

Totally Gnarly!

Reducing Radicals

Prime Factorization – the expression of a number as powers of its unique prime factors.

180EX(2)(90)

(2)(2)(45)(2)(2)(9)(5)

(2)(2)(3)(3)(5)2 22 3 5

There must be a factor of 2 in any even number.

45 is not prime so break it down.

Careful 9 can become (3)(3).

Finally combine all repeat primes.

Reducing Radicals

1715EX 2

(5)(343)Remember any number ending in 5 or 0 is divisible by 5.

343 hmm . . . Its not even and it doesn’t end in 5 or 0.

Is it divisible by 3?

Divisibility by 3 Test

A number is divisible by 3 if and only if the sum of its digits is divisible by 3.

EX 278154

2 7 8 1 5 4 27

Is divisible by 3?

Since 27 is divisible by 3, 278154 is also divisible by 3.

Reducing Radicals

Is 343 divisible by 3?

(5)(343)

(5)(7)(7)(7)35 7

Great! And 49 can becomes (7)(7).

Finally combine all repeat primes.

NO

It’s not divisible by 2, 3, or 5, let’s try 7.

(5)(7)(49)

Reducing Radicals

Ok, so what about those radicals!

EX 180 1.) Prime factorize the radicand.

The number under the radical

2 22 3 5 2.) Convert to fractional exponent form

1/22 22 3 5

Reducing Radicals

1/22 22 3 5 3.) Distribute the fractional exponent.

2(1/2) 2(1/2) 1/22 3 5 4.) Simplify and rewrite any ½ powers in radical form.

2 3 55.) Simplify

6 5

Reducing Radicals

EX 72592x

7(2)(1296)x

7(2)(2)(648)x

7(2)(2)(2)(324)x

7(2)(2)(2)(2)(162)x7(2)(2)(2)(2)(2)(81)x

5 72 (81)x

5 72 (3)(27)x

5 72 (3)(3)(3)(3)x

5 4 72 3 x

1/25 4 72 3 x2 2 32 3 2x x

336 2x x

Adding Radicals

Radicals are like fractions, only common radicals can be added or subtracted.

EX 2 5 3 5

5 5

EX 2 5 3 7

2 5 3 7Nothing can be done because the radicals do not have common radicands!

Adding Radicals

2 6 2 18 2 2 EX

Wait isn’t this impossible!

HA HA! Great!NO!

Adding Radicals

2 6 2 18 2 2 EXFirst we must reduce each radical!

22 2 3 2 2 3 2 2

2 6 2 3 2 2 2

2 6 8 2

2 6 6 2 2 2

Prime Factorize!

Can’t Simplify

Combine Common Radicals

Multiplying Radicals

Radicals are again like fractions, you multiply the matching parts.

3 2 6

5 7 35

5 2 3 7

Multiply the coefficients together.

Multiply the radicands together.

15 14

Multiplying Radicals

A monomial times a binomial

EX 25 3 7x x x

215x27 5x x Don’t forget to simplify

all radicals completely!

215 7 5x x x Unlike Radicands CANNOT be added.

Multiplying Radical

A binomial times a binomialEX 3 2 5 7 5 YOU MUST FOIL!!!!!

3 7 3 5 2 35 2 25

3 7 3 5 3 35 2(5)

3 7 3 5 3 35 10

Multiplying Radicals

EX 5 2 4 4 2 1

20 4 5 2 16 2 4

20(2) 5 2 16 2 4

40 21 2 4

44 21 2