PROPERTIES OF SQUARE ROOTS WORKSHEET

Question 1-6 : Simplify.

Question 1 :

√2 ⋅ √6

Question 2 :

5√6 ⋅ 3√8

Question 3 :

√32 ÷ √8

Question 4 :

5√8 ÷ 2√2

Question 5 :

3√425 + 4√68

Question 6 :

√243 - 5√12 + √27

Question 7 :

If √x = ½, then find the value of x.

Question 8 :

If (√9)⁵ ⋅ (√3)^-6 = 3^y, then solve for y.

Question 9 :

Solve for x :

x² = 25

Question 10 :

Solve for y :

3y² - 4 = 104

Answers

1. Answer :

= √2 ⋅ √6

= √(2 ⋅ 6)

= √(2 ⋅ 2 ⋅ 3)

= 2√3

2. Answer :

= 5√6 ⋅ 3√8

= (5 ⋅ 3)(√6 ⋅ √8)

= 15√(6 ⋅ 8)

= 15√(2 ⋅ 3 ⋅ 2 ⋅ 2 ⋅ 2)

= 15[2 ⋅ 2 ⋅ √3]

= 15(4√3)

= 60√3

3. Answer :

= √32 ÷ √8

= √(³²⁄₈)

= √4

= √(2 ⋅ 2)

= 2

4. Answer :

= 5√8 ÷ 2√2

= (⁵⁄₂)√(⁸⁄₂)

= (⁵⁄₂)(√4)

= (⁵⁄₂)(2)

= 5

5. Answer :

Decompose 425 and 68 into prime factors using synthetic division. 

3√425 + 4√68 : 

= 3(5√17) + 4(2√17)

= 15√17 + 8√17

= 23√17

6. Answer :

Decompose 243, 12 and 27 into prime factors using synthetic division. 

√243 = √(3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3) = 9√3

√12 = √(2 ⋅ 2 ⋅ 3) = 2√3

√27 = √(3 ⋅ 3 ⋅ 3) = 3√3

√243 - 5√12 + √27 : 

= 9√3 - 5(2√3) + 3√3

= 9√3 - 10√3 + 3√3

= 2√3

7. Answer :

√x = ½

Take square on both sides.

(√x)² = (½)²

x = 1²/2²

x = ¼

8. Answer :

 (√9)⁵ ⋅ (√3)^-6 = 3^y

3⁵ ⋅ (3^½)^-6 = 3^y

3⁵ ⋅ 3^-3 = 3^y

3^{5 - 3} = 3^y

3² = 3^y

y = 2

9. Answer :

x² = 25

Take square root on both sides.

√x² = ±√25

x = ±√(5 ⋅ 5)

x = ±5

x = -5  or  x = 5

10. Answer :

3y² - 4 = 104

Add 4 to both sides.

3y² = 108

Divide both sides by 3.

y² = 36

Take square root on both sides.

√y² = ±√36

y = ±√(6 ⋅ 6)

y = ±6

y = -6  or  y = 6

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