Simplify three under root three upon 2+2 under root two upon 3+5 under root five upon six
Answers
Answer:
Just as with "regular" numbers, square roots can be added together. But you might not be able to simplify the addition all the way down to one number. Just as "you can't add apples and oranges", so also you cannot combine "unlike" radical terms. In order to be able to combine radical terms together, those terms have to have the same radical part.
Simplify: 2 3 +3 3 \mathbf{\color{green}{ 2\,\sqrt{3\,} + 3\,\sqrt{3\,} }}23
+33
Since the radical is the same in each term (being the square root of three), then these are "like" terms. This means that I can combine the terms.
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Adding and Subtracting Square Roots
I have two copies of the radical, added to another three copies. This gives mea total of five copies:
2 3 +3 3 =(2+3) 3 2\,\sqrt{3\,} + 3\,\sqrt{3\,} = (2 + 3)\,\sqrt{3\,}23
+33
=(2+3)3
=5 3 = \mathbf{\color{purple}{ 5\,\sqrt{3\,} }}=53
That middle step, with the parentheses, shows the reasoning that justifies the final answer. You probably won't ever need to "show" this step, but it's what should be going through your mind.
Simplify: 3 +4 3 \mathbf{\color{green}{ \sqrt{3\,} + 4\,\sqrt{3\,} }}3
+43
The radical part is the same in each term, so I can do this addition. To help me keep track that the first term means "one copy of the square root of three", I'll insert the "understood" "1":
3 +4 3 =1 3 +4 3 \sqrt{3\,} + 4\,\sqrt{3\,} = 1\,\sqrt{3\,} + 4\,\sqrt{3\,}3
+43
=13
+43
=(1+4) 3 = (1 + 4)\,\sqrt{3\,}=(1+4)3
=5 3 = \mathbf{\color{purple}{ 5\,\sqrt{3\,} }}=53
Don't assume that expressions with unlike radicals cannot be simplified. It is possible that, after simplifying the radicals, the expression can indeed be simplified.
Simplify: 9 +25 \mathbf{\color{green}{\sqrt{9\,} + \sqrt{25\,}}}9
+25
To simplify a radical addition, I must first see if I can simplify each radical term. In this particular case, the square roots simplify "completely" (that is, down to whole numbers):
9 +25 =3+5=8\sqrt{9\,} + \sqrt{25\,} = 3 + 5 = \mathbf{\color{purple}{ 8 }}9
+25
=3+5=8
Simplify: 3 4 +2 4 \mathbf{\color{green}{ 3\,\sqrt{4\,} + 2\,\sqrt{4\,} }}34
+24
I have three copies of the radical, plus another two copies, giving me— Wait a minute! I can simplify those radicals right down to whole numbers:
3 4 +2 4 =3×2+2×23\,\sqrt{4\,} + 2\,\sqrt{4\,} = 3\times 2 + 2\times 234
+24
=3×2+2×2
=6+4=10= 6 + 4 = \mathbf{\color{purple}{ 10 }}=6+4=10
Don't worry if you don't see a simplification right away. If I hadn't noticed until the end that the radical simplified, my steps would have been different, but my final answer would have been the same:
3 4 +2 4 =5 4 3\,\sqrt{4\,} + 2\,\sqrt{4\,} = 5\,\sqrt{4\,}34
+24
=54
=5×2=10= 5\times 2 = 10=5×2=10
Simplify: 3 3 +2 5 +3 \mathbf{\color{green}{ 3\,\sqrt{3\,} + 2\,\sqrt{5\,} + \sqrt{3\,} }}33
+25
+3
I can only combine the "like" radicals. The first and last terms contain the square root of three, so they can be combined; the middle term contains the square root of five, so it cannot be combined with the others. So, in this case, I'll end up with two terms in my answer.
I'll start by rearranging the terms, to put the "like" terms together, and by inserting the "understood" 1 into the second square-root-of-three term:
3 3 +2 5 +3 3\,\sqrt{3\,} + 2\,\sqrt{5\,} + \color{red}{\sqrt{3\,}}33
+25
+3
=3 3 +1 3 +2 5 = 3\,\sqrt{3\,} + \color{red}{1\,\sqrt{3\,}} + 2\,\sqrt{5\,}=33
+13
+25
=4 3 +2 5 = \mathbf{\color{purple}{ 4\,\sqrt{3\,} + 2\,\sqrt{5\,} }}=43
+25
There is not, to my knowledge, any preferred ordering of terms in this sort of expression, so the expression 2 5 +4 3 2\,\sqrt{5\,} + 4\,\sqrt{3\,}25
+43
should also be an acceptable answer.
Step-by-step explanation: