What is x2+9x2+9? Many students will answer quickly that the answer is (x+3)(x+3) and have a very difficult time believing this answer is wrong. But it is wrong.
x2x2 is x*x*
and 99
is 3, but x2+9x2+9
is not (x+3)(x+3).
Why not? Remember that x2+9x2+9
is asking a question: “what squared gives the answer x2+9x2+9
?” So (x+3)(x+3) is not an answer, because (x+3)2
=x2+6x+9(x+3)2=x2+6x+9, not x2+9x2+9
.
As an example, suppose x=4x=4. So x2+9
=
42+9
=
25
=5x2+9=42+9=25=5. But (x+3)=7(x+3)=7.
If two numbers are added or subtracted under a square root, you cannot split them up. In symbols: a+b≠a+ba+b≠a+b
or, to put it another way,
x2
+
y2
≠a+b
x2
+
y2
≠a+b
x2+9x2+9
cannot, in fact, be simplified at all. It is a perfectly valid function, but cannot be rewritten in a simpler form.
How about 9x29x2
? By analogy to the previous discussion, you might expect that this cannot be simplified either. But in fact, it can be simplified:
9x2 =3x9x2=3x
Why? Again, 9x29x2 is asking “what squared gives the answer 9x2 9x2 ?” The answer is 3x3x because (3x)2=9x2 (3x)2=9x2.
Similarly, 9x2
=3x9x2=3x, because
3x2=9x23x2=9x2 size 12{ left ( { {3} over {x} } right ) rSup { size 8{2} } = { {9} over {x rSup { size 8{2} } } } } {}.
If two numbers are multiplied or divided under a square root, you can split them up. In symbols: ab = a bab = ab, ab =
abab= ab
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