Download Section 3.3 Square Roots Definition of a Square Root We know 5

Section 3.3 ­ Square Roots
Definition of a Square Root
­ We know 52 means: ­ Going backwards is finding the square root: "What number to the second power is 25?"
­ Let a be any positive real number. 2
Then a = b implies b
= a, √
and b is called the principal or positive square root of a.
Ex. 16 is asking "what number to the second power is 16?" √
Since 42 = 16, 16 = 4, so 4 is the principal square root. √
We can also write the negative square root as ­ 16 = ­4, since √
2
(­4) = 16.
Ex.1 ­ Evaluate each of the following. If a root is not a real number, explain why.
√
a.) 36
b.) ­ 100
√
c.) ­144
√
1
d.) √
Multiplying Radicals
­ If a ≥ 0 and b ≥ 0, then a b = a b The product of two √ √
√
square roots is equal to the square root of the product of the radicands.
Ex.2 ­ For each problem, multiply the radicals and evaluate.
√
a.) 2 18
√
b.) ­ 3 27
√
√
c.) √
5 5
√
Simplifying Radicals
­ If a ≥ 0 and b ≥ 0, then a b = a b A radical can √
√
√
be simplified by writing it as the product of two square roots.
Ex.3 ­ Simplify each of the following square roots exactly. Verify your results.
√
a.) 12 b.) 45
√
c.) √
32
d.) √
75
e.) √
200
More Simplifying
Ex.4 ­ Simplify. Give exact answers and approximate answers rounded to two decimal places.
√
a.) 3 2 4 26
√
b.) ­5 3 2 15
√
√
c.) √
(­9)2 ­ (2)(4)(7)
d.) √ 42 ­ (4)(­17)(4)
Rational Numbers and Irrational Numbers
­ Make up the set of real numbers (all numbers on the number line)
­ Rational numbers ­ a real number whose decimal expansion terminates or repeats (fractions, repeating or terminating decimals, integers). Examples: 0 or 0.14 or 0.3 or 52
­ Irrational numbers ­ a real number whose decimal expansion does not terminate and does not repeat (cannot be written as a fraction). Examples: 3 or 5 or √
√
π Ex.5 ­ Identify the rational numbers, the irrational numbers, and any non­real numbers.
1
√
­ 25 ­2.4 3 2 ­81 ­123 12
√
√
√
3