About 5 months ago, we discussed estimating square roots (EstSquareRoot). About 4 months ago, we calculated them (SquareRoot). Today we will simplify square roots. This concept appeared 5 times last year on middle school tests, with a median placement at question # 67.
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Number Dojo Level: 244
These questions require a fair amount of mental manipulation. For each one, there will be a square root that will not simplify completely (in other words, it will not be a perfect square). But at least one factor of the number under the radicand (square root sign) will be a perfect square. Our task is to factor out that perfect square and end up with the answer in its simplest form.
How to Simplify Square Roots:
- Factor the number under the radicand until you have a perfect square multiplied by some other number.
- Take the square root of that perfect square as you pull it out of (and place it in front of) the radicand. Leave the other factor inside.
- Repeat Steps 1 and 2 if possible.
Example 1: If √45 simplifies as a√b, then a = ___
- Factor 45 to 9 x 5.
- Take the square root of 9 to get 3. Place the 3 in front of the radicand and the 5 inside. You have 3√5.
- 5 does not have any factors which are perfect squares.
- The question asked for a, which is 3.
Example 2: If √98 simplifies as a√b, then a = ___
- 98 can be factored to 49 x 2.
- √49 is 7. Place this in front of the radicand and the 2 inside. You have 7√2.
- 2 doesn't have any other factors which are perfect squares.
- The question asked for a, which is 7.
Example 3: If √(4 x 8 x 12) = r√s, then s = ___
- (4 x 8 x 12) is the same as (32 x 2 x 6). This is (64 x 6).
- √64 = 8. Put this in front to make 8√6.
- √6 doesn't simplify any further. The answer is 6.
Example 4: √12 + √75 = √x. Find x.
- √12 simplifies to 2√3.
- √75 simplifies to 5√3.
- 2√3 + 5√3 = 7√3.
- Since the question asked for everything to be under the radicand, square 7 to get 49. Multiply this by 3 to get 147. The answer is 147.
Example 5: If √(5 x 10 x 25) simplifies to r√s, then r + s = ___
- √(5 x 10 x 25) = √(50 x 25) = √(2 x 25 x 25). This simplifies to 25√2.
- The question asked for r + s. 25 + 2 = 27.