1. Radicals NCP 503: Work with numerical factors
NCP 505: Work with squares and square roots of numbers
NCP 507: Work with cubes and cube roots of numbers
Radicals

2. What is a radical?
Square roots, cube roots, fourth roots, etc are all radicals.
They are the opposite of exponents.
√4 is asking what number times itself is equal to 4. (Answer is 2)
3√8 is asking what number times itself “3” times is equal to 8.
(Yup…the answer is 2 again.)

3. Square & Cube Roots Square Root Cube Root 3√125 √16 5 ∙ 5 ∙ 5 = 125
So, 3√125 = 5
4 ∙ 4 = 16
So, √16 = 4
Now you know what square and cube roots are, you can figure out the others…fourth root, fifth root, etc.

4. How to find a cube root! Look for the x√ , to enter 3√125
Enter 3, x√, 125, and EXE.

5. Try these…
3√1000 =
3√512 =
3√64 =
4√81 = 10 8 4 3

6. The square root of a perfect square is whole number.
Perfect Squares
2 ∙ 2 = 4, so 4 is a perfect square. Other perfect squares…
The square root of a perfect square is whole number.
√144 = 12

7. Not So Perfect Squares √50 √32 √20
Finding the square roots of other numbers results in a decimal. WE DO NOT WANT DECIMALS. NO DECIMALS! These will all end up as decimals. Remember: NO DECIMALS!
√50
√32
√20

8. Simplifying Square Roots
√8 =
√4 ∙ √2 =
2√2
8 is not a perfect square, so we will simplify it!
√4 = 2
We can’t simplify √2, so we leave him alone.
8 is made up of 4 ∙ 2. Look! 4 is a perfect square!
√50 =
√25 ∙ √2 =
5√2

9. Try these…
√27 √32 √20 √75
= √9 ∙ √3
= √16 ∙ √2
= √4 ∙ √5
= √25 ∙ √3
= 3√3
= 4√2
= 2√5
= 5√3

10. Combining Square Roots
To combine square roots, combine the coefficients of like square roots.
4√3 + 5 √3
= 9√3
They both have √3 in common, so we can add their coefficients.
They both have √3 in common, so we can add their coefficients.
7√2 – 4√2 =
3√2
Works with subtraction also.

11. Try these…
3√5 + 5√5
5√7 – 8√7
-2√3 + 7√3
7√11 – 4√11
√6 + 2√6
= 8√5
= -3√7
= 5√3
= 3√11
= 3√6

12. Combining Square Roots
We can combine multiple square roots!
6√3 + 4√5 – 2√3 + 2√5
= 4√3
+ 6√5
Combine the √3.
Combine the √3.
Next, combine the √5.
Next, combine the √5.
4√7 – 5√2 + 3√2 – 2√7 =
2√7 – 2√2

13. Try these…
-2√5 + 3√7 + 5√5
5√2 – 8√3 + 2√3
-4√6 + 2√5 – 3√6 + √5
√2 – 4√3 – 7√2 – √3
= 3√5 + 3√7
= 5√2 – 6√3
= -7√6 + 3√5
= -6√2 – 5√3

14. Simplify and Combine
√20 + √5 =
√4 ∙ √5 + √5 =
2√5 + √5 =
3√5
√12 + √27 =
√3 ∙ √4 +
√9 ∙ √3 =
2√3 +
3√3 =
5√3

15. Multiplying Radicals √3 ∙ √2 = √6 √3 ∙ √3 = √9 = 3 √3 ∙ √6 = √18
When multiplying radicals, you can multiply the two numbers and put the answer under one radical. Simplify!
√3 ∙ √2 = √6
√3 ∙ √3 = √9
= 3
√3 ∙ √6 =
√18
= √9 ∙ √2
= 3√2

16. Try This…
√7 ∙ √7 =
√49
= 7
√3 ∙ √5 =
√15
√2 ∙ √6 =
√12
= √4 ∙ √3
= 2√3
√15 ∙ √3 =
√45
= √9 ∙ √5
= 3√5

17. Let’s see these two examples!
Multiplying Radicals
When multiplying radicals, you must multiply the coefficients AND the radicals. THE RADICALS DO NOT HAVE TO BE THE SAME!
2√5 ∙ 3√5
4√2 ∙ 2√8
Let’s see these two examples!

18. 2√5 ∙ 3√5 Multiplying Radicals 1. Multiply the coefficients. 2 ∙ 3 = 6
2. Multiply the radicals.
√5 ∙ √5 = √25
3. Solve.
6√25 = 6 ∙ 5 = 30

19. 4√2 ∙ 2√3 Multiplying Radicals 1. Multiply the coefficients. 4 ∙ 2 = 8
2. Multiply the radicals.
√2 ∙ √3 = √6
3. Simplify, if possible.
8√6

20. Try This…
3√7 ∙ 2√5 =
6√35
2√3 ∙ 5√3 =
10√9 =
10 ∙ 3
= 30
4√2 ∙ 3√8 =
12√16
= 12 ∙ 4
= 48
2√5 ∙ 3√2 =
6√10