for each indicated sum and/or difference,simplify
the radicals and combine when possible.
That's too many to post. Here are three of them.
You do the rest. They're similar:
__ ___
5xÖ72 + Ö8x²
Break 72 into prime factors
72 = 8·9 = 2·4·3·3 = 2·2·2·3·3
Break 8x² into prime factors
8x² = 2·4·x·x = 2·2·2·x·x
Substitute under radicals:
_________ _________
5xÖ2·2·2·3·3 + Ö2·2·2·x·x
Group like factors into pairs:
_____________ _____________
5xÖ(2·2)·2·(3·3) + Ö(2·2)·2·(x·x)
Write each pair as a square:
_______ _______
5xÖ2²·2·3² + Ö2²·2·x²
Take the squares out in front of the radicals
as non-square factors:
_ _
5x·2·3Ö2 + 2·xÖ2
_ _
30xÖ2 + 2xÖ2
_
Factor out xÖ2
_
xÖ2(30 + 2)
_
xÖ2(32)
_
32xÖ2
===========================================
__ __ __ __
(1/3)Ö45 - (1/2)Ö12 + Ö20 + (2/3)Ö27
Break 45 into prime factors
45 = 9·5 = 3·3·5
Break 12 into prime factors
12 = 4·3 = 2·2·3
Break 20 into prime factors
20 = 4·5 = 2·2·5
Break 27 into prime factors
27 = 9·3 = 3·3·3
Substitute under radicals
_____ _____ _____ _____
(1/3)Ö3·3·5 - (1/2)Ö2·2·3 + Ö2·2·5 + (2/3)Ö3·3·3
Group like factors into pairs:
_______ _______ _______ _______
(1/3)Ö(3·3)·5 - (1/2)Ö(2·2)·3 + Ö(2·2)·5 + (2/3)Ö(3·3)·3
Write each pair as a square:
____ ____ ____ ____
(1/3)Ö3²·5 - (1/2)Ö2²·3 + Ö2²·5 + (2/3)Ö3²·3
Take the squares out in front of the radicals
as non-square factors:
_ _ _ _
(1/3)·3Ö5 - (1/2)·2Ö3 + 2Ö5 + (2/3)·3Ö3
_ _ _ _
Ö5 - Ö3 + 2Ö5 + 2Ö3
Group like radical terms together
_ _ _ _
Ö5 + 2Ö5 - Ö3 + 2Ö3
_ _
Factor Ö5 out of first two terms and Ö3 out of
last two terms:
_ _
Ö5(1 + 2) + Ö3(-1 + 2)
_ _
Ö5(3) + Ö3(1)
_ _
3Ö5 + Ö3
==============================================
______ _______
aÖ5000b³ + 2Ö125a³b²
Break 5000b³ into prime factors:
5000b³ = 2·2500·b·b·b = 2·2·1250·b·b·b = 2·2·2·625·b·b·b =
= 2·2·2·5·125·b·b = 2·2·2·5·5·25·b·b·b =
= 2·2·2·5·5·5·5·b·b·b
Break 125a³b² into prime factors
125a³b² = 5·25·a·a·a·b·b = 5·5·5·a·a·a·b·b
Substitute under radicals:
___________________ _______________
aÖ2·2·2·5·5·5·5·b·b·b + 2Ö5·5·5·a·a·a·b·b
Group like factor into pairs:
___________________________ _____________________
aÖ(2·2)·2·(5·5)·(5·5)·(b·b)·b + 2Ö(5·5)·5·(a·a)·a·(b·b)
Write each pair as a square:
_______________ ____________
aÖ2²·2·5²·5²·b²·b + 2Ö5²·5·a²·a·b²
Take the squares out in front of the radicals
as non-square factors:
___ ___
a·2·5·5·bÖ2·b + 2·5·a·bÖ5·a
__ __
50abÖ2b + 10abÖ5a
Factor out 10ab
__ __
10ab(5Ö2b + Ö5a)
Edwin