SIMPLIFYING RADICAL TERMS

Simplify :

Problem 1 :

(3√2)²

Solution :

(3√2)²  =  3√2 × 3√2

=  (3 ⋅ 3) [√(2 ⋅ 2)]

=  9 × 2

=  18

So, the answer is 18.

Problem 2 :

- 2√3 × 4√3

Solution :

-2√3 × 4√3  =  (- 2 ⋅ 4) [√(3 ⋅ 3)]

=  - 8 × 3

=  - 24

So, the answer is - 24.

Problem 3 :

3√2 - √8

Solution :

=  3√2 - √8

By decomposing √8, we get

=  3√2 - √(4  2)

=  3√2 - 2√2

=  √2 

So, the answer is √2.

Problem 4 :

 2√3 × 3√5

Solution :

2√3 × 3√5  =  (2 ⋅ 3) [√(3 ⋅ 5)]

=  6√15

So, the answer is 6√15.

Problem 5 :

(2√5)²

Solution :

(2√5)²  =  2√5 × 2√5

=  (2 ⋅ 2) [√(5 ⋅ 5)]

=  4 × 5

=  20

So, the answer is 20.

Problem 6 :

5√2 - 7√2

Solution :

=  5√2 - 7√2

=  - 2√2

So, the answer is - 2√2.

Problem 7 :

(√3)⁴

Solution :

(√3)⁴  =  √3 ⋅ √3 ⋅ √3 ⋅ √3

=  (√3 ⋅ 3) (√3 ⋅ 3)

=  3 × 3

=  9

So, the answer is 9.

Problem 8 :

 √3 × √5 × √15

Solution :

=  √3 × √5 × √15

=  √(3 ⋅ 5 ⋅ 15)

=  √(15 ⋅ 15)

=  15

So, the answer is 15.

Problem 9 :

Write √48 in simplest radical form.

Solution :

√48  =  √(2 ⋅ 2 ⋅ 3)

=  2√3

So, the simplest radical form is 4√3

Problem 10 :

Write √75 in simplest radical form.

Solution :

 √75  =  √(5 ⋅ 5 ⋅3)

=  5√3

So, the answer is 5√3.

Problem 11 :

What is the set of all solutions to the equation

√(x + 18) = −2x?

(A) {−2, −6}     (B) {−2}    (C) {3}

(D) There are no solutions to the given equation.

Solution :

√(x + 18) = −2x

Squaring on both sides, we get

x + 18 = (-2x)²

x + 18 = 4x² 

Subtracting x and 18 on both sides

4x² - x - 18 = 0

4x² - 9x + 8x - 18 = 0

x(4x - 9) + 2(4x - 9) = 0

(x + 2)(4x - 9) = 0

Equating each factor to 0, we get

x + 2 = 0 and 4x - 9 = 0

x = -2 and x = 9/4

So, the values of x are -2 and 9/4. Then option B is correct.

Problem 12 :

³√8x³

If x < 0, which of the following is equivalent to the expression above ?

(A) -8x     (B) -2x   (C) 2x    (D) 2x³

Solution :

= ³√8x³

= ³√2⋅2⋅2⋅x⋅x⋅x

Since it is cube root, we have to factor out only one value for every three same values.

=  2x

Problem 13 :

√x = x - 2

What is the greatest value of x that satisfies the above equation ?

Solution :

√x = x - 2

Squaring on both sides, 

(√x)² = (x - 2)²

x = x²- 4x + 4

x²- 4x - x + 4 = 0

x²- 5x + 4 = 0

(x - 1)(x - 4) = 0

x = 1 and x = 4

So, the greatest value of x is 4.

Problem 14 :

If a = 3√7/4 and 4a = √3b, what is the value of b ?

(A) 4/3     (B) 7   (C) 21    (D) 63

Solution :

a = 3√7/4 ------(1)

4a = √3b -----(2)

From (1), multiplying by 4 on both sides, we get

4a = 3√7

From (2), we have the value of 4a.

3√7 = √3b

Squaring on both sides

(3√7)² = (√3b)²

9(7) = 3b

3b = 63

b = 63/3

b = 21

So, the value of b is 21.

Problem 15 :

Simplify the radical expression 6/√3 ?

Solution :

6/√3

Since we have radical at the denominator, in order to remove the radical, we have to multiply by /√3 in both numerator and denominator.

= (6/√3) ⋅ (√3/√3)

= 6√3/3

= 2√3

So, the answer is 2√3.

Problem 16 :

2√45 - 2√5

Solution :

2√45 - 2√5

= 2√(3⋅3⋅5) - 2√5

= (2⋅3)√5 - 2√5

= 6√5 - 2√5

= 4√5

Problem 17 :

-3√5 - √6 - √5

Solution :

= -3√5 - √6 - √5

By combing the like terms, we get

= -4√5 - √6

Problem 18 :

√2(√6 - √10)

Solution :

= √2(√6 - √10)

Using distributive property, 

= √2 √6 - √2√10

= √(2x6) - √(2x10)

= √(2x2x3) - √(2x2x5)

= 2√3 - 2√5

We cannot combine, because these two are not like terms.

So, the answer is 2√3 - 2√5.

Problem 19 :

√49a⁸x¹²

Solution :

= √49a⁸x¹²

Trying to decompose each term as product of two same terms. 

= √(7 ⋅ 7 ⋅ a⁴ ⋅ a⁴⋅ x⁶ ⋅ x⁶)

Since we have two same terms which are multiplied inside the square root sign, we have to factor one value out of the square root sign.

= 7 a⁴x⁶

Problem 20 :

√5xy √40x³y

Solution :

= √5xy √40x³y

Using the properties of square root,

= √(5xy ⋅ 40x³y)

= √(5xy ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5 x³y)

= 5 ⋅ 2 x² y √2

= 10x²y √2

So, the answer is 10x²y √2.

Problem 21 :

³√162x⁵/³√3x²

Solution :

= ³√162x⁵/³√3x²

Using the properties of square root,

= ³√(162x⁵/ 3x²)

= ³√(54 x³)

= ³√(3 ⋅ 3⋅ 3 ⋅ 2 x³)

= 3 x ³√2

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