radical operations
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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