Topic 1.3.2
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1 Topic 1.3.2 Square Roots
description
Square Roots. Topic 1.3.2. Topic 1.3.2. Square Roots. California Standard: 2.0: Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. - PowerPoint PPT Presentation
Transcript of Topic 1.3.2
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Topic 1.3.2Topic 1.3.2
Square RootsSquare Roots
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Topic1.3.2
Square RootsSquare Roots
California Standard:2.0: Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.
What it means for you:You’ll look more closely at the rules of square roots.
Key words:• square root• radical• radicand• principal square root• minor square root
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Topic1.3.2
Square RootsSquare Roots
In the last Topic you learned about all the exponent rules — this Topic will look more closely at one rule in particular.
Square roots are the type of root that you’ll come across most often in math problems — so it’s really important that you know how to deal with them.
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Topic1.3.2
Another Name for the Root Sign is the Radical Sign
Square RootsSquare Roots
The square root of p is written .
If you multiply by itself, you get p — so × = p.
Multiplying by itself means you square it.
If you raise to the power n, you get p — so = p.n
The symbol is called the radical sign and shows the nonnegative root if more than one root exists. In the expression (the nth root of p), p is the radicand.
The nth root of p is written .
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Topic1.3.2
Square RootsSquare Roots
The square root of a number p is also written p .12
For any real number p > 0, the square root is written as (or p ).
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If r = , then r2 = p and (–r)2 = p.
r is called the principal square root of p and –r is called the minor square root of p.
You can show this using the rules of exponents:
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22 1p = p = p = p
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2. The 6th root of t is written in radical notation. ……..
Topic1.3.2
Guided Practice
Solution follows…
Square RootsSquare Roots
Complete the following.
1. The radicand of 8 is .……..3
3. 9 × = 9……..
4. b = in radical notation.12
……..
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t6
9
b
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Topic1.3.2
Positive Numbers Have Two Square Roots
Square RootsSquare Roots
Every positive number has two square roots — a positive one (the principal square root) and a negative one (the minor square root).
The principal square root of n is written as . The minor square root of n is written as – . To indicate both square roots you can write ± .
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Topic1.3.2
Example 1
Solution follows…
Square RootsSquare Roots
Find the square roots of the following numbers:
a) 100 b) n2
Solution
a) 100 = 10So the principal square root is 10, and the minor square root is –10.
b) n2 = |n|
So the principal square root is |n|, and the minor square root is –|n|.
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Topic1.3.2
Guided Practice
Solution follows…
Square RootsSquare Roots
Find the principal square root and minor square root of these numbers:
5. 4 6. 100 7. 81
Use the “±” symbol to give the principal and minor square root of the following numbers:
8. 9 9. 16 10. 144
11. 352 12. x2 13. 81
14. t2 15. 9 × 9 16. (st)2
Evaluate the following, giving the principal and minor roots:
17. 412 18. 121
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2 and –2 10 and –10 9 and –9
±3 ±4 ±12
±35 ±x ±9
±t ±9 ±st
±2 ±11
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Topic1.3.2
Algebraic Expressions Also Have Square Roots
Square RootsSquare Roots
You can also take the square root of an algebraic expression.
Remember — algebraic expressions contain variables, which represent unknown values.
For example, a + b or 2t4.
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Topic1.3.2
Example 2
Solution follows…
Square RootsSquare Roots
Find the square root of (x + 1)2.
Solution
(x + 1)2 = |x + 1|
So the principal square root is |x + 1| and the minor square root is –|x + 1|.
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Topic1.3.2
Guided Practice
Solution follows…
Square RootsSquare Roots
Give the principal and minor square root of each of the following expressions.
19. t × t ±t 20. t2 × t2 ±t2
21. a2 × a2 ±a2 22. (a + b) × (a + b) ±(a + b)
23. t(a + b) × t(a + b) ±t(a + b) 24. (a + b)2 ±(a + b)
25. (t + 1)2 ±(t + 1) 26. [t(a + b)]2 ±t(a + b)
27. [2(a + b)]2 ±2(a + b)
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Topic1.3.2
Independent Practice
Solution follows…
Square RootsSquare Roots
1. Is this statement true or false? “The radicand of 32 is 5.”5
Evaluate the following.
2. 64123. (49) 4. a2
5. 25 6. 122 7. j × j
Find the square roots of the following.
8. (a2)2 9. (k – 1)2 10. (m + n)2
11. (m2 + n2)2 12. (2pq)2 13. [(a + b) × (c + d)]2
False. The radicand is 32.
8 7 |a|
5 12 |j|
±a2 ±(k – 1) ±(m + n)
±(m2 + n2) ±2pq ±(a + b)(c + d)
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Topic1.3.2
Round UpRound Up
Square RootsSquare Roots
Remember that when you take the square root of a positive number, you always have two possible answers — a positive one and a negative one.
You can give both answers neatly using the ± sign.
