Download MATH 5: HOMEWORK 7 Square Root Question: What is the square

MATH 5: HOMEWORK 7
MARCH 3, 2013
Square Root
Question: What is the square of a number, say 2? Answer: The square of 2 is 4 because
2 × 2 = 4.
Exercise: What is the square of 5?
√
The square root of a number, say 4 is 2 because 2 × 2 = 4. We write it as: 4 = 2.
Note that also (−2) × (−2) = 4, but for the moment we will be considering only positive
solutions.
√
In general the square root of a number a2 is a; we can write it as a2 = a.
To easily find the squre root of a number, write the prime factorization for the number
and group them as (...)2 . For example:
√
484 = 2 × 2 × 11 × 11 = 22 × 112 = (2 × 11)2 So 484 = 2 × 11 = 22.
The square of a Sum
We learned how to compute using the distributive property:
(a + b)(x + y) = a(x + y) + b(x + y) = ax + ay + bx + by
What happens when we want to multiply the same sum?
(a + b)2 = (a + b)(a + b) = a(a + b) + b(a + b) = aa + ab + ba + bb = a2 + 2ab + b2
(a + b)2 = a2 + 2ab + b2
(1)
Example: What is (x + 1)2 ? We can use the formula and save time!
(x + 1)2 = x2 + 2x + 1
Exercise: Compute (x + 2)2 =
The square of a Difference
Similarly when multiplying the same difference with itself:
(a − b)2 = (a − b)(a − b) = a(a − b) − b(a − b) = aa − ab − ba + bb = a2 − 2ab + b2
(2)
Exercise: Compute (x − 3)2 =
(a − b)2 = a2 − 2ab + b2
The difference of Squares
(a + b)(a − b) = a(a − b) + b(a − b) = aa − ab + ba − bb = a2 − b2
Let’s write this result starting with the rightmost side:
a2 − b2 = (a − b)(a + b)
(3)
In other words, you can always write a difference of 2 squares as a product of 2 factors.
Note that 12 = 1, so x2 − 12 = x2 − 1 = (x − 1)(x + 1).
Exercise: Factor 4x2 − 1 =
Homework
1. Write the squares of all consecutive numbers from 1 to 20.
√
√
√
√
2. What is 144, 169, 100, 676?
3. Compute:
(a)(2x)2 =
√
(d) 121 × 169 =
(b)(4xy)2 =
√
(e) 22 × 34 =
(c)(5y 4 )2 =
√
(f) 10000 =
(Remember the rule (ab)m = am bm when we talked about exponents, you have to raise
to the power each factor. For example: (3abc)2 = 32 a2 b2 c2 = 9a2 b2 c2 )
4. Compute using difference of squares:
(a)142 − 42 =
(b)262 − 42 =
(c)1212 − 192 =
5. What is:
(a)(x − 5)2 =
(b)(x + 2)2 =
(c)y 2 − x2 =
6. Solve the Math Kangaroo problems in the given handout.