Download Math I Unit 5 Lesson 2 Investigation 5 – Square Roots and

Math I
Unit 5 Lesson 2
Investigation 5 – Square Roots and Radicals
Name: ___________________________________
In your work on problems of insulin decay, you found that some questions required calculation with exponential
expressions involving a fractional base and fractional powers. For example, estimating the amount of insulin
active in the bloodstream 1.5 minutes after a 10-unit injection required calculating 10(0.951.5 ).
Among the most useful expressions with fractional exponents are those with power one-half. It turns
out that one-half powers are connected to the square roots that are so useful in geometric calculations like those
involving the Pythagorean Theorem. For any non-negative number b.
1
𝑏 2 = √𝑏
Expressions like √𝑏, √5, and √9 − 𝑥 2 are called radicals. As you work on the following problems, keep this
question in mind:
How can you use your understanding of properties of exponents to guide your thinking about one-half
powers, square roots, radical expressions, and rules for operating with them?
1. For integer exponents m and n, you know that (𝑎𝑚 )𝑛 = 𝑎𝑚𝑛 . That property can be extended to work with
fractional exponents.
a. Write each of these expressions in standard number form without exponents or radicals.
1⁄ 2
2)
1⁄ 2
2)
i. (2
1⁄ 2
2)
ii. (5
iii. (12
b. How do the results of Part a explain why the definition 𝑏
1⁄
2
1⁄ 2
2)
iv. (2.4
= √𝑏 makes sense?
2. Write each of the following expressions in an equivalent form using radicals and then in simplest number
form (without exponents or radicals).
a. (25)
1⁄
2
b. (9)
1⁄
2
9
1⁄
2
c. (4)
d. (100)
1⁄
2
3. The diagram below shows a series of squares with side lengths increasing in sequence 1, 2, 3, 4, and one
diagonal drawn in each square.
1
2
3
4
a. Use the Pythagorean Theorem (𝑎2 + 𝑏 2 = 𝑐 2 ) to find the exact length of the diagonal of each square.
b. How are the lengths of the diagonals in the three larger squares related to the length of the diagonal
of the unit square?
c. Look for a pattern in the results of Part b to complete the statement beginning:
The length d of each diagonal in a square with sides of length s is given by d = ______________
4. The pattern relating side and diagonal lengths in a square illustrates a useful rule for simplifying radical
expressions:
For any non-negative numbers a and b: √𝑎𝑏 = √𝑎√𝑏.
a. What properties of square roots and exponents justify the steps in this argument? For any nonegative numbers a and b:
1
(1)
√𝑎𝑏 = (𝑎𝑏) ⁄2
=𝑎
1⁄ 1⁄
2𝑏 2
(2)
= √𝑎√𝑏
(3)
b. Modify the argument in Part a to justify this property of radicals:
𝑎
For any non-negative numbers a and b (𝑏 ≠ 0), √𝑏 =
√𝑎
√𝑏
.
5. Use the properties of square roots in Problem 4 to write expressions a-h in several equivalent forms. In each
case, try to find the simplest equivalent form – one that involves only one radical and the smallest possible
number inside that radical. Check your ideas with calculator estimates of each form. For example,
√48 = √4√12 = 2√12 = 2√4√3 = 2 ∙ 2√3 = 4√3
Calculator estimates show that √48 ≈ 6.93 and 4√3 ≈ 6.93.
a. √9 ∙ 5
1
e. √4 ∙ 9
b. √18√8
c. √45
d. √4 ∙ 9
9
g. √12
h. √96
f. √4
6. The properties of square roots in Problem 4 are like distributive properties – taking the square root
distributes over the product or the quotient of two (or more) numbers. One of the most common errors in
working with square roots is distributing the square root sign over addition. However, √𝑎 + 𝑏 ≠ √𝑎 + √𝑏
except in some very special cases. Use several pairs of positive values for a and b to show that taking square
roots does not distribute over addition (or subtraction).