**Square Roots and Cube Roots**

**Square Roots:**

**Definition: The square root of a number is a value that, when multiplied by itself, gives the original number.**

**Simplification Rule: A square root can be simplified if it has a perfect square factor in the radicand.**

**For example, √12 can be simplified as √(4 x 3) = 2√3.**

**Multiplication Rule: The product of two square roots is equal to the square root of their product.**

**For example, √a x √b = √(ab).**

**Division Rule: The quotient of two square roots is equal to the square root of their quotient.**

**For example, √a ÷ √b = √(a/b).**

**Cube Roots:**

**Definition: The cube root of a number is a value that, when multiplied by itself three times, gives the original number.**

**Simplification Rule: A cube root can be simplified if it has a perfect cube factor in the radicand.**

**For example, ∛54 can be simplified as ∛(27 x 2) = 3∛2.**

**Multiplication Rule: The product of two cube roots is equal to the cube root of their product.**

**For example, ∛a x ∛b = ∛(ab).**

**Division Rule: The quotient of two cube roots is equal to the cube root of their quotient.**

**For example, ∛a ÷ ∛b = ∛(a/b).**

**Practice Questions on Square Roots Simplification Rule**

**Question 1:**

**Simplify √(16) + √(9).**

**Solution:**

**√(16) = 4 (since the square root of 16 is 4)**

**√(9) = 3 (since the square root of 9 is 3)**

**Therefore, √(16) + √(9) = 4 + 3 = 7.**

**Requested by: Shahbaz**

**Question 2:**

**Simplify √(25) – √(4).**

**Solution:**

**√(25) = 5 (since the square root of 25 is 5)**

**√(4) = 2 (since the square root of 4 is 2)**

**Therefore, √(25) – √(4) = 5 – 2 = 3.**

**Requested by: Shahbaz**

**Question 3:**

**Simplify √(49) + √(64) – √(36).**

**Solution:**

**√(49) = 7 (since the square root of 49 is 7)**

**√(64) = 8 (since the square root of 64 is 8)**

**√(36) = 6 (since the square root of 36 is 6)**

**Therefore, √(49) + √(64) – √(36) = 7 + 8 – 6 = 9.**

**Requested by: Shahbaz**

**Question 4:**

**Simplify √(121) – √(144) + √(169).**

**Solution:**

**√(121) = 11 (since the square root of 121 is 11)**

**√(144) = 12 (since the square root of 144 is 12)**

**√(169) = 13 (since the square root of 169 is 13)**

**Therefore, √(121) – √(144) + √(169) = 11 – 12 + 13 = 12.**

**Requested by: Shahbaz**

**Question 5:**

**Simplify √(36) + √(81) – √(64) + √(49).**

**Solution:**

**√(36) = 6 (since the square root of 36 is 6)**

**√(81) = 9 (since the square root of 81 is 9)**

**√(64) = 8 (since the square root of 64 is 8)**

**√(49) = 7 (since the square root of 49 is 7)**

**Therefore, √(36) + √(81) – √(64) + √(49) = 6 + 9 – 8 + 7 = 14.**

**Requested by: Shahbaz**

**Question 6:**

**Simplify √144.**

**Solution:**

**√144 = √(12^2) = 12**

**Requested by: Shahbaz**

**Question 7:**

**Simplify √(8√2).**

**Solution:**

**√(8√2) = √(2 × 4√2) = √(2 × 2√(2^2)) = √(2 × 2√4) = √(2 × 2 × 2) = √(8) = 2√2**

**Requested by: Shahbaz**

**Question 8:**

**Simplify √(√(81) + √(16)).**

**Solution:**

**√(√(81) + √(16)) = √(√(9^2) + √(4^2)) = √(√(9^2) + √(4^2)) = √(√9^2 + √4^2) = √(√9 + √4) = √(3 + 2) = √5**

**Requested by: Shahbaz**

**Question 9:**

**Simplify √(√(27) – √(12)).**

**Solution:**

**√(√(27) – √(12)) = √(√(3^3) – √(2^2 × 3)) = √(√3^3 – √2^2 × √3) = √(√3 × 3 – √4 × √3) = √(3√3 – 2√3) = √(√3) = √3**

**Requested by: Shahbaz**

**Practice Questions on Square Roots Multiplication Rules**

**Question 1:**

**Simplify √7 * √5.**

**Solution:**

**√7 * √5 = √(7 * 5) = √35**

**Requested by: Shahbaz**

**Question 2:**

**Simplify 2√3 * 3√2.**

**Solution:**

**2√3 * 3√2 = 6√(3 * 2) = 6√6**

**Requested by: Shahbaz**

**Question 3:**

**Simplify (√6 + √2) * (√6 – √2).**

**Solution:**

**(√6 + √2) * (√6 – √2) = (√6)^2 – (√2)^2 = 6 – 2 = 4**

**Requested by: Shahbaz**

**Question 4:**

**Simplify (2√3 – √5) * (3√3 + √5).**

**Solution:**

**(2√3 – √5) * (3√3 + √5) = (2√3)^2 – (√5)^2 = 4 * 3 – 5 = 12 – 5 = 7**

**Requested by: Shahbaz**

**Question 5:**

**Simplify (√2 + √3)(√2 – √3).**

**Solution:**

**(√2 + √3)(√2 – √3) = (√2)^2 – (√3)^2 = 2 – 3 = -1**

**Requested by: Shahbaz**

**Question 6:**

**Simplify √8 * √18.**

**Solution:**

**√8 * √18 = √(8 * 18) = √144 = 12**

**Requested by: Shahbaz**

**Question 7:**

**Simplify (√5 – √2)(√5 + √2).**

**Solution:**

**(√5 – √2)(√5 + √2) = (√5)^2 – (√2)^2 = 5 – 2 = 3**

**Requested by: Shahbaz**

**Question 8:**

**Simplify (√10 + 3)(√10 – 3).**

**Solution:**

**(√10 + 3)(√10 – 3) = (√10)^2 – 3^2 = 10 – 9 = 1**

**Requested by: Shahbaz**

**Question 9:**

**Simplify (√7 + 2√3)(√7 – 2√3).**

**Solution:**

**(√7 + 2√3)(√7 – 2√3) = (√7)^2 – (2√3)^2 = 7 – 12 = -5**

**Requested by: Shahbaz**

**Practice Questions on Square Roots Division Rule**

**Question 1:**

**Simplify √20 / √5.**

**Solution:**

**√20 / √5 = (√(4 × 5)) / √5 = (√4 × √5) / √5 = 2**

**Requested by: Shahbaz**

**Question 2:**

**Simplify √27 / √3.**

**Solution:**

**√27 / √3 = (√(9 × 3)) / √3 = (√9 × √3) / √3 = 3**

**Requested by: Shahbaz**

**Question 3:**

**Simplify (√18 – √2) / (√2 – √6).**

**Solution:**

**(√18 – √2) / (√2 – √6) = [(√(9 × 2)) – (√2)] / [(√2) – (√(3 × 2))] = [(√9 × √2) – (√2)] / [(√2) – (√3 × √2)] = [3√2 – √2] / [√2 – √(3 × 2)] = 2**

**Requested by: Shahbaz**

**Question 4:**

**Simplify (√15 + √5) / (√3 + √5).**

**Solution:**

**(√15 + √5) / (√3 + √5) = [(√(3 × 5)) + √5] / [(√3) + √5] = [(√3 × √5) + √5] / [√3 + √5] = [√15 + √5] / [√3 + √5] = 1**

**Requested by: Shahbaz**

**Question 5:**

**Simplify (√8 – √2) / (√2 + √6).**

**Solution:**

**(√8 – √2) / (√2 + √6) = [(√(4 × 2)) – (√2)] / [(√2) + (√(3 × 2))] = [(√4 × √2) – (√2)] / [√2 + √(3 × 2)] = [2√2 – √2] / [√2 + √(3 × 2)] = √2**

**Requested by: Shahbaz**

**Practice Questions on Cube Roots Simplification Rule**

**Question 1:**

**Simplify ∛64.**

**Solution:**

**∛64 = 4**

**Requested by: Shahbaz**

**Question 2:**

**Simplify ∛(-125).**

**Solution:**

**∛(-125) = -5**

**Requested by: Shahbaz**

**Question 3:**

**Simplify ∛(2∛(27)).**

**Solution:**

**∛(2∛(27)) = ∛(2∛(3^3)) = ∛(2∛27) = ∛(2∛(3^3)) = ∛(2∛3^3) = ∛(2∛27) = ∛(2∛(3^3)) = ∛(2∛3^3) = ∛(2 * 3) = ∛6**

**Requested by: Shahbaz**

**Question 4:**

**Simplify ∛(∛(64) + ∛(8)).**

**Solution:**

**∛(∛64 + ∛8) = ∛(∛(4^3) + ∛(2^3)) = ∛(∛4^3 + ∛2^3) = ∛(∛4 + ∛2) = ∛(∛(2^2) + ∛(2)) = ∛(2 + ∛2) = ∛(2 + 1) = ∛3**

**Requested by: Shahbaz**

**Question 5:**

**Simplify ∛(∛(27) – ∛(125)).**

**Solution:**

**∛(∛27 – ∛125) = ∛(∛(3^3) – ∛(5^3)) = ∛(∛3^3 – ∛5^3) = ∛(∛3 – ∛5) = ∛(∛(3^2) – ∛(5)) = ∛(∛9 – ∛5) = ∛(2 – ∛5)**

**Requested by: Shahbaz**

**Practice Questions on Cube Roots Multiplication Rule**

**Question 1:**

**Simplify ∛8 * ∛4.**

**Solution:**

**∛8 * ∛4 = ∛(2^3) * ∛(2^2) = 2 * 2 = 4**

**Requested by: Shahbaz**

**Question 2:**

**Simplify ∛(3∛27).**

**Solution:**

**∛(3∛27) = ∛(3∛(3^3)) = ∛(3 * 3) = ∛9 = 2**

**Requested by: Shahbaz**

**Question 3:**

**Simplify ∛5 * ∛4.**

**Solution:**

**∛5 * ∛4 = ∛(5 * 4) = ∛20**

**Requested by: Shahbaz**

**Question 4:**

**Simplify ∛(-3) * ∛(-27).**

**Solution:**

**∛(-3) * ∛(-27) = ∛(-3 * -27) = ∛81 = 3**

**Requested by: Shahbaz**

**Question 5:**

**Simplify ∛(2∛(3)) * ∛(3∛(2)).**

**Solution:**

**∛(2∛(3)) * ∛(3∛(2)) = ∛(2 * 3) * ∛(3 * 2) = ∛6 * ∛6 = ∛(6 * 6) = ∛36 = 6**

**Requested by: Shahbaz**

**Question 6:**

**Simplify ∛(∛(8) * ∛(27)).**

**Solution:**

**∛(∛8 * ∛27) = ∛(2 * 3) = ∛(6) = 6**

**Requested by: Shahbaz**

**Question 7:**

**Simplify (∛5 + ∛3) * (∛5 – ∛3).**

**Solution:**

**(∛5 + ∛3) * (∛5 – ∛3) = (∛5)^2 – (∛3)^2 = 5 – 3 = 2**

**Requested by: Shahbaz**

**Practice Questions on Cube Roots Division Rule**

**Question 1:**

**Simplify ∛8 / ∛2.**

**Solution:**

**∛8 / ∛2 = ∛(8 / 2) = ∛4 = 2**

**Requested by: Shahbaz**

**Question 2:**

**Simplify ∛(27) / ∛(3).**

**Solution:**

**∛27 / ∛3 = (∛(3^3)) / (∛3) = (∛3)^3 / (∛3) = 3**

**Requested by: Shahbaz**

**Question 3:**

**Simplify ∛8 / ∛2.**

**Solution:**

**∛8 / ∛2 = ∛(8 / 2) = ∛4 = 2**

**Requested by: Shahbaz**

**Question 4:**

**Simplify ∛27 / ∛9.**

**Solution:**

**∛27 / ∛9 = ∛(27 / 9) = ∛3 = 1.442**

**Requested by: Shahbaz**

**Question 5:**

**Simplify (∛125 – ∛25) / ∛5.**

**Solution:**

**(∛125 – ∛25) / ∛5 = (∛(5^3) – ∛(5^2)) / ∛5 = (∛5 – ∛5) / ∛5 = 0**

**Requested by: Shahbaz**

**Question 6:**

**Simplify ∛(∛64 / ∛16).**

**Solution:**

**∛(∛64 / ∛16) = ∛(∛(4^3) / ∛(4^2)) = ∛(∛4 / ∛4) = ∛(1) = 1**

**Requested by: Shahbaz**

**Question 7:**

**Simplify ∛(∛216 / ∛8).**

**Solution:**

**∛(∛216 / ∛8) = ∛(∛(6^3) / ∛(2^3)) = ∛(∛6 / ∛2) = ∛(∛3) = 1.442**

**Requested by: Shahbaz**