Radicals & Rational Exponents — Drills
Radicals & Rational Exponents has 6 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills 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.
Simplifying
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Simplify √360.
Worked answer
6sqrt(10)
√36 · 10
= 6√10
Check: 62 × 10 = 360 ✓
Problem 2
Simplify √448.
Worked answer
8sqrt(7)
√64 · 7
= 8√7
Check: 82 × 7 = 448 ✓
Exponents
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Write x4/4 as a radical.
Worked answer
4th root of x^4
√[4]x4
Problem 2
Write √[5]x3 using a rational exponent.
Worked answer
x^(3/5)
x3/5
Evaluating
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Evaluate 253/2.
Worked answer
125
√[2]25 = 5
53 = 125
The other order would mean √[2]15625 — same answer, far more work.
Problem 2
Evaluate 1253/3.
Worked answer
125
√[3]125 = 5
53 = 125
The other order would mean √[3]1953125 — same answer, far more work.
Adding
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Simplify √125 + √80.
Worked answer
9sqrt(5)
√125 = 5√5
√80 = 4√5
5√5 + 4√5 = 9√5
Problem 2
Simplify √112 + √7.
Worked answer
5sqrt(7)
√112 = 4√7
√7 = 1√7
4√7 + 1√7 = 5√7
Multiplying
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Simplify √6 · √6.
Worked answer
6
√6 × 6 = √36
= 6
Problem 2
Simplify √7 · √5.
Worked answer
sqrt(35)
√7 × 5 = √35
No square factor, so that is simplest form.
Irrational
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Is 5 + √5 rational or irrational?
Worked answer
irrational
√5 is irrational and 5 is rational.
A rational plus an irrational is always irrational.
Problem 2
Is √224 · √14 rational or irrational?
Worked answer
rational
√3136
= 56
Rational — two irrationals can multiply to a rational.