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

NCP 503: Work with numerical factorsNCP 505: Work with squares and square roots of numbersNCP 507: Work with cubes and cube roots of numbers

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.)

Square & Cube Roots

√16 4 ∙ 4 = 16So, √16 = 4

3√125 5 ∙ 5 ∙ 5 = 125So, 3√125 = 5

Square Root Cube Root

Now you know what square and cube roots are, you can figure out the others…fourth root, fifth root, etc.

How to find a cube root!

Look for the x√ , to enter 3√125Enter 3, x√, 125, and EXE.

Try these…

3√1000 =

3√512 =

3√64 =

4√81 =

10

8

4

3

Perfect Squares2 ∙ 2 = 4, so 4 is a perfect square.

Other perfect squares…9 81 22516 100 25625 121 28936 144 32449 169 36164 196 400

The square root of a perfect

square is whole number.

√144 = 12

Not So Perfect SquaresFinding 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

Simplifying Square Roots

√8 = √4 ∙ √2 = 2√2

8 is not a perfect

square, so we will

simplify it!

8 is made up of 4 ∙ 2. Look! 4 is a

perfect square!

√4 = 2We can’t

simplify √2, so we leave him alone.

√50 =√25 ∙ √2 = 5√2

Try these…

√27

√32

√20

√75

= √9 ∙ √3

= √16 ∙ √2

= √4 ∙ √5

= √25 ∙ √3

= 3√3

= 4√2

= 2√5

= 5√3

Combining Square Roots

To combine square roots, combine the coefficients of like square roots.

4√3 + 5 √3= 9√3

7√2 – 4√2 =

They both have √3 in common, so we can add their coefficients.

They both have √3 in common, so we can add their coefficients.

3√2

Works with subtraction also.

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

Combining Square Roots

We can combine multiple square roots!

6√3 + 4√5 – 2√3 + 2√5 = 4√3

4√7 – 5√2 + 3√2 – 2√7 =

Next, combine the √5. Combine the √3.

2√7 – 2√2

+ 6√5

Combine the √3. Next, combine the

√5.

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

Simplify and Combine

√20 + √5 =√4 ∙ √5 + √5

=

√12 + √27 =

2√3

+√3 ∙ √43√3 =

√9 ∙ √3 =

5√3

2√5 + √5 =

3√5

+

Multiplying RadicalsWhen 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

Try This…

√7 ∙ √7 =

√49

√3 ∙ √5 =

√15

= 7

√2 ∙ √6 =

√12 = √4 ∙ √3

= 2√3

√15 ∙ √3 =

√45 = √9 ∙ √5

= 3√5

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!

Multiplying Radicals

2√5 ∙ 3√5

1. Multiply the coefficients.2 ∙ 3 = 6

2. Multiply the radicals.√5 ∙ √5 = √25

3. Solve.6√25 = 6 ∙ 5 = 30

Multiplying Radicals

4√2 ∙ 2√3

1. Multiply the coefficients.4 ∙ 2 = 8

2. Multiply the radicals.√2 ∙ √3 = √6

3. Simplify, if possible.8√6

Try This…

3√7 ∙ 2√5 =6√35

2√3 ∙ 5√3 =10√9 =10 ∙ 3

4√2 ∙ 3√8 =12√16= 12 ∙ 4

= 48

2√5 ∙ 3√2 =6√10

= 30