SIMPLIFYING VARIABLE EXPRESSIONS WITH SQUARE ROOTS

Simplify each radical expression. Use absolute value symbols when needed.

Problem 1 :

√16x²

Solution :

√16x² = √ (4 ∙ 4 ∙ x ∙ x)

√16x² = 4x

Problem 2 :

√0.25x⁶

Solution :

√0.25x⁶ = √ (0.5) (0.5) . x^{3 .} x³

√0.25x⁶ = 0.5 x³

Problem 3 :

√x⁸y¹⁸

Solution :

√x⁸y¹⁸ = √ (x⁴ ∙ x⁴ ∙ y⁹ ∙ y⁹)

√x⁸y¹⁸ = x⁴ ∙ x⁹

= x⁴⁺⁹

= x¹³

Problem 4:

√64b⁴⁸

Solution :

√64b⁴⁸ = √ (8 ∙ 8 ∙ b²⁴ ∙ b²⁴)

√64b⁴⁸ = 8 b²⁴

Problem 5 :

-√64a³

Solution :

-√64a³ = (4 ∙ 4 ∙ 4 ∙ a ∙ a ∙ a)

-√64a³ = - 4a

Problem 6 :

√27y⁶

Solution :

√27y⁶ = (3 ∙ 3 ∙ 3 ∙ y² ∙ y² ∙ y²)

√27y⁶ = 3y²

Problem 7 :

√x⁸ y¹²

Solution :

√x⁸y¹² = (x² ∙ x² ∙ x² ∙ x² ∙ y³ ∙ y³ ∙ y³∙ y ³)

√x⁸y¹² = x²y³

Problem 8 :

√x¹⁰

Solution :

√x^{10 =} √x⁵  x⁵

= x⁵

Problem 9 :

√y²

Solution :

√y² = √(y ∙ y)

√y² = y

Problem 10 :

√b²

Solution :

√b² = √ (b ∙ b)

√b² = b

Problem 11 :

√49 x²

Solution :

√49x² = √ (7 ∙ 7 ∙ x ∙ x)

√49x² = 7x

Problem 12 :

√100 y²

Solution :

√100 y² = √ (10 ∙ 10 ∙ y ∙ y)

√100y² = 10y

Problem 13 :

-√64 a²

Solution :

-√64 a² = -√ (8 ∙ 8 ∙ a ∙ a)

-√64 a² = - 8 a 

Problem 14 :

-√25 x²

Solution :

-√25 x² = -√(5 ∙ 5 ∙ x ∙ x)

-√25 x² = - 5 x

Problem 15 :

√144 x² y²

Solution :

√144 x² y² = √(12 ∙ 12 ∙ x ∙ x ∙ y ∙ y)

√144 x² y² = 12 x y

Problem 16 :

√196a²b²

Solution :

√196a²b² = √ (14 ∙ 14 ∙ a ∙ a ∙ b ∙ b)

√196 a² b² = 14 a b

Problem 17 :

Which of the following is equal to x ?

a) x^12/7 - x^5/7      b)  ¹²√ (x⁴)^1/3         c)  (√x²)^2/3     d) x^12/7 ∙ x^7/12

Solution :

Option a :

Since we have negative sign in the middle, we cannot simplify more.

Option b :

= ¹²√ (x⁴)^1/3 

= ¹²√ (x^4 ∙1/3)

= ¹²√ (x^4/3)

= (x^4/3)^1/12

= x^{(4/3) ∙ (1/12)}

= x^{(1/3) ∙ (1/3)}

= x^1/9

It is not equal to x.

Option c :

= (√x²)^2/3   

= ((x²)^1/2)^2/3

= x^2/3

Option d :

= x^12/7 ∙ x^7/12

=  x^{(12/7) ∙ (7/12)}

= x

So, option d is equal to x.

Problem 18 :

Find the value of x in 3 + 2^x = 64^1/2 + 27^1/3

Solution :

3 + 2^x = 64^1/2 + 27^1/3

3 + 2^x = (8²)^1/2 + (3³)^1/3

3 + 2^x = 8^{2 ∙ (1/2)} + 3^{3 ∙ (1/3)}

3 + 2^x = 8 + 3

2^x = 8 + 3 - 3

2^x = 8

2^x = 2³

x = 3

So, the value of x is 3.

Problem 19 :

If x = (√3 - √2) / (√3 + √2) and y = (√3 + √2) / (√3 - √2) find the value of x² + y² + xy

Solution :

x = (√3 - √2) / (√3 + √2)

x = [(√3 - √2) / (√3 + √2)] ∙ [(√3 - √2) / (√3 - √2)]

x = (√3 - √2)² / (√3 + √2)(√3 - √2)

x = (√3² - 2√(3x2) + √2²) / (√3² - √2²)

x = (3 - 2√6 + 2) / (3 - 2)

x = (5 - 2√6) / 1

x = (5 - 2√6)

x² = (5 - 2√6)²

= 5² - 2(5)(2√6) + (2√6)²

= 25 - 20√6 + 24

x²  = 49 - 20√6

y = (√3 - √2) / (√3 + √2)

y = [(√3 + √2) / (√3 - √2)] ∙ [(√3 + √2) / (√3 + √2)]

x = (√3 + √2)² / (√3 + √2)(√3 - √2)

y = (√3² + 2√(3x2) + √2²) / (√3² - √2²)

y = (3 + 2√6 + 2) / (3 - 2)

y = (5 + 2√6) / 1

y = (5 + 2√6)

y² =  (5 + 2√6)²

= 5² + 2(5)(2√6) + (2√6)²

= 25 + 20√6 + 24

y²  = 49 + 20√6

xy = (√3 - √2) / (√3 + √2) (√3 + √2) / (√3 - √2)

xy = 1

x² + y² + xy = 49 - 20√6 + 49 + 20√6 + 1

= 49 + 49 + 1

= 99

So, the value of x² + y² + xy is 99.

Problem 20 :

Solve for x in the following questions.

i)  √(32.4/x) = 2      ii)  √86.49 + √(5 + x²) = 12.3

Solution :

i)  √(32.4/x) = 2

To remove the square root sign, squaring on both sides.

32.4/x = 2²

32.4 / x = 4

x = 32.4/4

x = 8.1

So, the missing value is 8.1

ii)  √86.49 + √(5 + x²) = 12.3

9.3 + √(5 + x²) = 12.3

√(5 + x²) = 12.3 - 9.3

√(5 + x²) = 3

(5 + x²) = 3²

(5 + x²) = 9

x² = 9 - 5

x² = 4

x = -2 and 2.