How to do Radical Expressions on the Calculator

Square Root √ = first push

 then push

 

Cube root ³√ = first push

then push 4

 

 

x root ^x√ = First push 

then push 5

7-2 Definitions

New Vocabulary: rationalize the denominator.

Definition 1: To rationalize the denominator of a function, rewrite it so there are no radicals in any denominator and no denominators in any radical.
Definition 2: You make sure that there is no square,cube, etc. roots in the denominator, and no fractions inside a square, cube, etc. root.

lesson 7-2 Normal

 

When multiplying radicals, you must multiply the numbers OUTSIDE (O) the radicals AND then multiply the numbers INSIDE (I) the radicals.

When dividing radicals, you must divide the numbers OUTSIDE (O) the radicals AND then divide the numbers INSIDE (I) the radicals.

lesson 7-2 Advance

     To multiply radicals, consider the following, √(25) * √(9) = 5 * 3 = 15 and √(25) * √(9) = √(225) = 15, so √(25) * √(9) = √(25*9).

     as a general rule, √(a) * √(b) = √(ab)


     To simplify radicals, such as √(72), where there is no perfect square, you need to take the square out, so it would be √(2*6²), which would simplify as 6√2


     To divide radicals, consider the following, √(25)/√(9) = 5/3 and √(25)/√(9) = √(25/9) = √(5/3)² = 5/3


     as a general rule, if a and b are real numbers, and b does not equal zero, √(a)/√(b) = √(a/b)


      To rationalize the denominator, such as √(5)/√(3), make the bottom a perfect square, like this, √(5*3)/√(3*3). I multiplied both the top and bottom by √(3)/√(3) to make the denominator a perfect square, making the problem look like √(15)/3, which is rationalized.

*note, a number with an exponent of 1/2 is equal to taking the square root of it, an exponent of 1/3 is the same as the cube root.*

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Basic problems

1. √16 * √9
2. -5^1/3 * 25^1/3
3. √4 * √25
4. 8^1/3 * 27^1/3
5. √3 * √12
6. 3^1/3 * 9^1/3
7. √36 / √25
8. 32^1/3 / -4^1/3
9. (4² )^1/4
10. √2 / √3

Advanced problems

1.  √(72x³) 
2.  (80n⁵)^⅓ 
3.  (54x²y^³)⅓ * (5x³y⁴)⅓ 
4.  3√(7x³) * 2√(21x³y²) 
5. (1024x¹⁵)^1/4 / (4x)^1/4 
6.  (2/3x)⅓ 
7.  √((x³)/(5xy)) 
8.  3(5y³)⅓ * 2(50y⁴)⅓ 
9.  √(48x³)/√(3xy²)
10.  E=mc² , solve for c and rationalize the denominator
    
    
    Assume all variables are positive

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answers

Beginner Level answers:

1. √16 * √9 = 4 * 3 = 12

2. ³√5 * ³√25 = ³√125 = 5

3. √4 * √25 = √100 = 10

4. ³√8 * ³√27 = 2*3 = 6

5. √3 * √12 = √36 = 6

6. ³√3 * ³√9 = ³√27 = 3

7. √36 / √25 = 6/5

8. ³√32 / ³√-4 = ³√64 / ³√-8 = -2

9. (4² )^{1/4 = 2}

10. √2 / √3 = √6 / 3

Advanced level answers

1. √(72x³) = √(6² * x²) * √(2 * x) = 6x√(2x)

2. ³√(80n⁵) = ³√(2³ * x³ * 10 * x²) = 2x ³√(10x²)

3. (54x²y^³)^⅓ * (5x³y⁴)^⅓ = 3y ³√(50x²y²) / 5xy²

4.  3√(7x³) * 2√(21x³y²) = 21x² √(3) / 42x²y

5. (1024x¹⁵)^1/4 / (4x)^{1/4 =} 4x^{4 4}√(x²) / x

6. (2/3x)⅓ = ³√(18x² ) / 3x

7. √((x³)/(5xy)) = x²√(5y) / 5xy

8. 3(5y³)⅓ * 2(50y⁴)⅓ = 6y ³√(1.5y²) / 10y²

9. √(48x³)/√(3xy²) = 4x/y

10. E = mc², c² = E/m, c = √(E/m), c = √(Em/m²), c = √(Em)/m

extra problems

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