1. Radicals 1

2. I am not a teacher; only a fellow traveler of whom you asked the way.
George Bernard Shaw ( )
British dramatist, critic, writer. 2

3. Squaring a Number
42 means 4 x 4 = 16. It is called 4 squared and 16 is called a square number.
A square with sides that are 4 long has an area of 16 units2
42 = 16
You can see the reason for calling it 4 squared and why 16 is a square number

4. Perfect Squares
The numbers 16, 36, and 49 are examples of perfect squares. A perfect square is a number that has integers as its base. Other perfect squares include 1, 4, 9, 25, 64, and 81.
Number 1 2 3 4 5
Square 1 4 9 16 25
Number 6 7 8 9 10
Square 36 49 64 81
100

5. Square Roots
Many mathematical operations have an inverse, or opposite, operation. Subtraction is the opposite of addition, division is the inverse of multiplication, and so on. Squaring has an inverse too, called "finding the square root."

6. Square Roots
Remember, the square of a number is that number times itself.
The square root of a number, n, written
is the number that gives n when multiplied by itself. For example,
because 10 x 10 = 100

7. Square Roots √16 means ‘the square root of 16’ and √16 = 4
A square with an area of 16 has sides that are 4 units long.
Taking the square root of a number is the reverse process of squaring the number.

8. Perfect square
Square root
Perfect square
Square root 1
Ö1 = 1 81
Ö81 = 9 4
Ö4 = 2
100
Ö100 = 10 9
Ö9 = 3
121
Ö121 = 11 16
Ö16 = 4
144
Ö144 = 12 25
Ö25 = 5
169
Ö169 = 13 36
Ö36 = 6
196
Ö196 = 14 49
Ö49 = 7
225
Ö225 = 15 64
Ö64 = 8

9. Square Roots since 4 × 4 = 16. The other square root of 16 is –4,
Every positive number has two square roots, one positive and one negative. One square root of 16 is 4,
since 4 × 4 = 16.
The other square root of 16 is –4,
since (–4) × (–4) is also 16.
You can write the square roots of 16 as ±4, meaning “plus or minus” 4.

10. Square Roots The radical symbol returns only the positive root.
This is called the principal square root of the number.
To get the negative root, simply take the opposite of the principal root:

11. What about negatives? Consider a negative under the radical
This is asking ‘what number multiplied by itself returns a -25?’
5×5 = 25
(-5) ×(-5) = 25
No product returns a negative value
positive
positive

12. Example
A square shaped kitchen table has an area of 16 square feet. Will it fit through a van door that has a 5 foot wide opening?
Find the square root of 16 to find the width of the table. Use the positive square root; a negative length has no meaning.
So the table is 4 feet wide, which is less than 5 feet, so it will fit through the van door.

13. Estimating Square Roots
Finding square roots of numbers that aren't perfect squares without a calculator
Estimate - first, get as close as you can by finding two perfect square roots your number is between.
Divide - divide your number by one of those square roots.
Average - take the average of the result of step 2 and the root.
Use the result of step 3 to repeat steps 2 and 3 until you have a number that is accurate enough for you.

14. 3 ? 4 Estimating Square Roots
Calculate the square root of 10 without a calculator
Estimate - first, get as close as you can by finding two perfect square roots your number is between. 3 ? 4
Lies between 3 and 4

15. 10 = 3.33 3 Estimating Square Roots Calculate the square root of 10
Divide - divide your number by one of those square roots. 10
= 3.33 3
Divide 10 by 3.
(you can round off your answer)

16. 3.33 + 3 = 3.1667 2 Estimating Square Roots
Calculate the square root of 10
Average - take the average of the result of step 2 and the root chosen in step 2. = 2
Average 3.33 and 3.

17. Estimating Square Roots
Calculate the square root of 10
Use the result of step 3 to repeat steps 2 and 3 until you have a number that is accurate enough for you. 10 =
Repeat step 2 with 10 and
3.1667 =
Repeat step 3 with
and 2

18. Estimating Square Roots
Calculate the square root of 10
Try the answer →
Is squared equal to 10?
x =
If this is accurate enough for you, you can stop! Otherwise, you can repeat steps 2 and 3.

19. Multiplying Radicals = 2 × 3 = 6 = 6 Consider the following product
Another way:
= 6

20. Rules of Radicals
This process leads to a few simple rules we can use with radicals

21. Multiply

22. Simplifying Radicals
There are three components to a simplified radical:
All perfect square factors should be removed from the radical
All fractions should be removed from the radical
All radicals should be removed from the denominator

23. Simplify Radicals
Recall we prefer to simplify fractions by removing common factors to make the denominator smaller.
For radicals, we will follow a similar process, except we will concentrate on perfect square factors
Perfect Square

24. Simplify Think of the largest perfect square that divides into 32
4 yes → 4×8
9 no
16 yes → 16×2

25. Simplify Think of the largest perfect square that divides into 32 4 no
9 yes → 9×5

26. Simplify
Even if you can’t think of the largest perfect square, you can always simplify down through each perfect square:

27. Simplify
Use the division property
Check:
2/3 ͯ 2/3 = 4/9

28. Simplify Use the division property to write as single fraction
Simplify the fraction
Use the division property again

29. Simplify How do we remove the radical from the denominator? Recall:
All radicals should be removed from the denominator
How do we remove the radical from the denominator?

30. Since this is a fraction, then let’s think about how we change the denominator of a fraction?
(Without changing the value of the fraction, of course.)

31. We multiply the denominator and the numerator by the same number
How can we change the radical value to a rational value?

32. Simply multiply the radical by itself!
Remember when we square a square root, the radical goes away

33. In our fraction, to get the radical out of the denominator, we can multiply numerator and denominator by

34. we call this process rationalizing.
Because we are changing the denominator to a rational number,
we call this process rationalizing.

35. Simplify
Use the division property
Rationalize the denominator

36. Exponents under the Radical
Variables with exponents may exist under the radical, but the same simplifying process still applies
All even exponents are perfect squares
Notice the pattern
half the exponent

37. Exponents under the Radical
To simplify, we can use the perfect square property
No more radical
Rewrite as a perfect square
half of 10 is 5
The radical and the square offset each other

38. Simplify
Half of the exponent
6 is half of 12
4 is half of 8

39. Exponents under the Radical
Odd exponents can be rewritten as an even and an odd by using the rules of exponents
Rewrite odd exponent as one less (even) plus one
3 = 2 + 1
9 = 8 + 1
15 =
101 =

40. Simplify Rewrite odd exponent as one less (even) and one
half of 6 is 3
the other x stays under the radical

41. Adding Radicals Adding radicals is same as adding like terms:
1 + 1 = 2
x + x = 2x
 +  = 2 

42. Adding and Subtracting Like Radicals
Add or subtract, as indicated. Assume all variables represent positive real numbers.
Cannot add yet. Simplify to see if they are like radicals

43. Combined Operations with Radicals
You follow the same steps with these as you do with polynomials.
Use the distribution property.
Example: