Master Radical Expressions: Conquer Radicals in Mathematics and Statistics

Question 1

Rationalize the denominator of: 1 / √5

Accepted Answer

√5 / 5

Answer

5 / √5

Answer

√5 / 25

Answer

25 / √5

Question 2

Simplify the expression: (√a)(√b)(√c)

Accepted Answer

√(abc)

Answer

√a + √b + √c

Answer

ab + bc + ca

Answer

a√bc

Question 3

Rationalize the numerator of: (√x + 1) / (√x - 1)

Accepted Answer

(√x + 1)(√x + 1) / (√x - 1)(√x + 1)

Answer

(√x - 1)(√x - 1) / (√x + 1)(√x - 1)

Answer

(√x + 1) / (√x - 1)

Question 4

Can you explain the main distinction between simplifying and rationalizing a radical expression?

Accepted Answer

Simplifying is about operating within the radical, while rationalizing is about operating outside it.

Answer

Simplifying purifies radicals, while rationalizing introduces them.

Answer

Simplifying is exclusive to even exponents, while rationalizing is exclusive to odd exponents.

Question 5

Simplify the radical expression: √(16x²)

Accepted Answer

4x

Answer

2x

Answer

16x

Answer

8x

Question 6

Solve for x in the radical equation: √(x + 5) = 3

Accepted Answer

4

Answer

16

Answer

1

Answer

9

Question 7

Simplify the expression: (√2 + √3)²

Accepted Answer

5 + 2√6

Answer

11

Answer

10

Answer

5 - 2√6

Question 8

Solve the radical equation: (x + 5)^1/2 = 3

Accepted Answer

4

Answer

1

Answer

16

Answer

9

Question 9

Simplify the radical expression: (8 - 2√7)^1/2

Accepted Answer

√7 - 1

Answer

√7 + 1

Answer

4 - √7

Answer

2 - √7

Question 10

Solve the radical inequality: 2(x - 1)^1/2 < 6

Accepted Answer

1 < x < 13

Answer

x > 13

Answer

x < 13

Answer

x < 1

Question 11

Simplify the radical expression: (27a^3b^6)^1/2

Accepted Answer

3a^2b^3

Answer

9a^2b^3

Answer

3a^3b^6

Answer

9a^3b^6

Question 12

Simplify the radical expression √(18x3y2):

Accepted Answer

3x√(2xy^2)

Answer

6x^2y^3

Answer

9x^2y^2

Answer

6xy

Question 13

Rationalize the denominator of the expression 1/(√2 - 1):

Accepted Answer

√2 + 1

Answer

√2 - 1

Answer

1/2

Answer

-√2 + 1

Question 14

Solve the radical equation √(x + 5) = 3:

Accepted Answer

4

Answer

-9

Answer

-4

Answer

9

Question 15

Which of the following is equivalent to (√3 + √5)(√3 - √5):

Accepted Answer

-2

Answer

4

Answer

10

Answer

8

Question 16

Simplify the radical expression √(24/5):

Accepted Answer

2√6/5

Answer

3√2/5

Answer

5√2/6

Answer

4√3/5

Question 17

Simplify the radical expression: √18.

Accepted Answer

3√2

Answer

9√2

Answer

6

Answer

√9√2

Question 18

Rationalize the denominator of the expression: 3/√5.

Accepted Answer

3√5/5

Answer

√5/3

Answer

√15/5

Answer

3/5

Question 19

Which of the following is equivalent to √(x^3) ?

Accepted Answer

x√x

Answer

x^(3/2)

Answer

x^6

Answer

x^(1/3)

Question 20

Simplify the radical expression: √(27/4).

Accepted Answer

3√3/2

Answer

√27/√4

Answer

9√3/2

Answer

3/2

Question 21

Solve for x: √(x + 5) = √(2x - 1).

Accepted Answer

x = 6

Answer

x = 1

Answer

x = 3

Answer

x = -6

Question 22

Simplify the expression: √(16a^4b^6).

Accepted Answer

4a^2b^3

Answer

2a^2b^3

Answer

8a^2b^3

Answer

4a^4b^6

Question 23

Rationalize the denominator of the expression: 3/√5.

Accepted Answer

3√5 / 5

Answer

√15 / 5

Answer

√5 / 3

Answer

3/5

Question 24

Rationalize the denominator of the expression: 2 / (√3 - √2).

Accepted Answer

2(√3 + √2)

Answer

2√3 + 2√2

Answer

(√3 + √2) / 2

Answer

2(√3 - √2)

Question 25

Solve for x: √(2x - 1) = √(x + 4).

Accepted Answer

x = 5

Answer

x = -5

Answer

x = -3

Answer

x = 3

Question 26

Simplify the radical expression: √72.

Accepted Answer

6√2

Answer

3√8

Answer

12√2

Answer

9√8

Question 27

Simplify the radical expression: √(x⁶y⁸)

Accepted Answer

x3y4

Answer

x6y4

Answer

x3y8

Answer

x9y12

Question 28

Rationalize the denominator of the expression: 3/√5

Accepted Answer

3√5 / 5

Answer

3√5 / √25

Answer

√5 / 3

Answer

9√5 / 5

Question 29

Simplify the expression: √12 + √27

Accepted Answer

5√3

Answer

6√3

Answer

√39

Answer

3√3

Question 30

What is the simplified form of √(4x²y⁴)?

Accepted Answer

2xy2

Answer

√(2xy2)

Answer

4xy2

Answer

2x2y2

Question 31

Solve for x in the equation: √(2x - 1) = √(x + 4)

Accepted Answer

x = 5

Answer

x = -1

Answer

x = 1

Answer

x = 3

Question 32

Simplify the expression: (√5 + √2)(√5 - √2)

Accepted Answer

3

Answer

√3

Answer

7

Answer

√7

Question 33

Solve the equation √(x + 3) = x - 1 for x. Remember to check for extraneous solutions.

Accepted Answer

x = 4

Answer

x = 2

Answer

x = -2

Answer

x = 1

Question 34

Simplify the radical expression: √(48)

Accepted Answer

4√3

Answer

12

Answer

6√2

Answer

2√12

Question 35

Solve the radical equation: √(x - 2) = 5

Accepted Answer

x = 27

Answer

x = 5

Answer

x = 12

Answer

x = 7

Question 36

Identify the conjugate of the expression: √2 - 3

Accepted Answer

√2 + 3

Answer

-√2 + 3

Answer

2√2 + 3

Answer

√2 - 3

Question 37

Which of the following expressions is irrational?

Accepted Answer

√3 + 2

Answer

4

Answer

0.5

Answer

1/2

Question 38

Find the domain of the expression: 1/√(x - 3)

Accepted Answer

x > 3

Answer

x >= 3

Answer

x < 3

Question 39

Simplify the expression √(16x^8y^10):

Accepted Answer

4x^4y^5

Answer

2x^4y^5

Answer

4x^8y^10

Answer

2x^8y^10

Question 40

Rationalize the denominator of the expression 1/√5:

Accepted Answer

√5/5

Answer

5

Answer

5/√5

Answer

1/5

Question 41

Solve for x in the equation √(x + 5) = 3:

Accepted Answer

4

Answer

25

Answer

9

Answer

16

Question 42

Which of the following is equivalent to (√3 + √2)^2?

Accepted Answer

5 + 2√6

Answer

5 - 2√6

Answer

11 - 2√6

Answer

11 + 2√6

Question 43

Simplify the expression √(12) - √(3):

Accepted Answer

√3

Answer

-√3

Answer

3

Answer

2√3

Question 44

Rationalize the numerator of the expression √(3)/√(2):

Accepted Answer

√6/2

Answer

√6

Answer

3√2/2

Answer

3√2

Question 45

Solve for x in the equation 2√(x - 1) = 6:

Accepted Answer

5

Answer

9

Answer

1

Answer

16

Question 46

Which of the following is equivalent to (√2 - √3)^2?

Accepted Answer

5 - 2√6

Answer

5 + 2√6

Answer

11 - 2√6

Question 47

Simplify the expression √(98a^3b^5):

Accepted Answer

7a√2b^2

Answer

14a^2√2b^3

Answer

14a√2b^2

Answer

7a^2√2b^3

Question 48

Rationalize the denominator of the expression 1/(√a + √b):

Accepted Answer

(√a - √b)/(a - b)

Answer

(a + b)/(√a + √b)

Answer

(a - b)/(√a - √b)

Answer

(√a + √b)/(a + b)

Question 49

Simplify the expression: √(75)

Accepted Answer

5√3

Answer

25√3

Answer

3√5

Question 50

Which of the following is equivalent to √(48x²y) assuming x and y are non-negative?

Accepted Answer

4x√(3y)

Answer

4√(3xy)

Answer

16x√(3y)

Answer

2x√(12y)

Question 51

Simplify: (√7 - √2)(√7 + √2)

Accepted Answer

5

Answer

√5

Answer

9

Answer

√14

Question 52

Rationalize the denominator: 5 / √3

Accepted Answer

(5√3) / 3

Answer

(5√3) / 9

Answer

5√3

Question 53

Rationalize the denominator: 2 / (√5 - 1)

Accepted Answer

(√5 + 1) / 2

Answer

(2√5 - 1) / 4

Answer

(2√5 - 2) / 4

Answer

(2√5 + 2) / 2

Question 54

Solve for x: √(x + 7) = 3

Accepted Answer

x = 2

Answer

x = 16

Answer

x = -4

Answer

x = 9

Question 55

Solve for x: √(2x - 1) - √(x - 4) = 1

Accepted Answer

x = 4, x = 13

Answer

x = 4

Answer

x = 13

Answer

x = 5

Question 56

What is the conjugate of the expression: 3 - √2 ?

Accepted Answer

3 + √2

Answer

-3 + √2

Answer

-3 - √2

Answer

3 - √2

Question 57

The area of a square is 50 square units. What is the length of one side of the square in simplified radical form?

Accepted Answer

5√2 units

Answer

25√2 units

Answer

√50 units

Answer

10√2 units

Question 58

Which of the following is an irrational number?

Accepted Answer

√7

Answer

√9

Answer

-5

Answer

3/4