Radicals & Rational Exponents — Practice
A root is a fractional power. Once you accept that, every exponent rule you know still applies.
What these questions cover
These questions use 7 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Exponent — How many times the base is used as a factor. In 53 the exponent is 3.
- Square root — For k ≥ 0, √k is the nonnegative number whose square is k.
- Rational — Any number writable as ab with integers a and b, b ≠ 0.
- Irrational — A real number that cannot be written as a fraction — its decimal never repeats and never ends.
- Factor — A number that divides evenly into another.
- Difference of squares — a2 - b2 = (a - b)(a + b).
- Rational exponent — x1/2 = √x and xm/n = √[n]xm.
Problem 1
Simplify √50.
Worked answer
5sqrt(2)
√25 · 2
= 5√2
Problem 2
Simplify √75.
Worked answer
5sqrt(3)
√25 · 3
= 5√3
Problem 3
Simplify √48.
Worked answer
4sqrt(3)
√16 · 3
= 4√3
Problem 4
Write √x using a rational exponent.
Worked answer
x^(1/2)
x1/2
Problem 5
Evaluate 161/2.
Worked answer
4
√16 = 4
Problem 6
Simplify √12 + √27.
Worked answer
5sqrt(3)
√12 = 2√3
√27 = 3√3
2√3 + 3√3 = 5√3
Problem 7
Simplify 3√5 - 7√5.
Worked answer
-4sqrt(5)
3 - 7 = -4
-4√5
Problem 8
Simplify √6 · √10.
Worked answer
2sqrt(15)
√60
= √4 · 15 = 2√15
Problem 9
Evaluate 272/3.
Worked answer
9
√[3]27 = 3
32 = 9
Problem 10
Evaluate 84/3.
Worked answer
16
√[3]8 = 2
24 = 16
Problem 11
Which sum is irrational?
- 3 + √2
- √4 + √9
- 12 + 13
- √16 - 1
Worked answer
3 + √2
√2 is irrational and 3 is rational
So the sum is irrational
All the others reduce to rationals
Problem 12
Write √[3]x5 using a rational exponent.
Worked answer
x^(5/3)
x5/3
Problem 13
Simplify √200 - √18.
Worked answer
7sqrt(2)
√200 = 10√2
√18 = 3√2
10√2 - 3√2 = 7√2
Problem 14
Expand (√3 + 2)(√3 - 2).
Worked answer
-1
(√3)2 - 22
= 3 - 4 = -1
A rational answer from irrational parts
Problem 15
Evaluate (116)-1/2.
Worked answer
4
(161)1/2
= √16 = 4
Problem 16
Is √8 · √2 rational or irrational?
Worked answer
rational
√16
= 4
Rational — two irrationals can multiply to a rational