Radicals & Rational Exponents — Challenge
A root is a fractional power. Once you accept that, every exponent rule you know still applies.
What these problems 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
Write √48 in simplest radical form.
- 4√3
- 6√2
- 3√6
- 16√3
- 2√6
Worked answer
4√3
Find the largest perfect square dividing 48: that is 16, since 48 = 16· 3.
√48 = √16·√3 = 4√3.
Using 4 instead would give 2√12, correct in value but not simplest, since 12 still contains the square 4.
Problem 2
Simplify √18+√2.
- √20
- 3√2
- 4√2
- 5√2
- 6√2
Worked answer
4√2
Simplify the first term: √18 = √9·√2 = 3√2.
Now both terms contain √2, so they can be added like terms: 3√2+√2 = 4√2.
Adding the numbers inside the signs to get √20 is wrong: roots do not add that way.
Problem 3
Simplify √8·√6.
- 4√3
- √14
- 48
- 2√3
- 12√2
Worked answer
4√3
Roots multiply under a single sign: √8·√6 = √48.
Then simplify: √48 = √16·√3 = 4√3.
Adding the insides to get √14 would be the wrong rule.
Problem 4
Rationalise the denominator of 1√2.
- √2
- √22
- 12
- 2√2
- √24
Worked answer
√22
Multiply the top and the bottom by √2, which does not change the value.
The bottom becomes √2·√2 = 2 and the top becomes √2.
So the answer is √22, about 0.707, the same value as the original.
Problem 5
Simplify (√5)2+√9.
- 8
- √14
- 15
- 25
- 6
Worked answer
8
Squaring a square root undoes it exactly: (√5)2 = 5.
√9 = 3 because 9 is a perfect square.
5+3 = 8.
Problem 6
Simplify √254.
- 254
- 54
- 52
- 25
- 5
Worked answer
52
The root of a fraction is the root of the top over the root of the bottom.
√25√4 = 52.
As a decimal that is 2.5, and 2.52 = 6.25 = 254, which checks the answer.
Problem 7
Write √12 in simplest radical form.
- 2√3
- 3√2
- 4√3
- 2√6
- 6√2
Worked answer
2√3
The largest perfect square dividing 12 is 4, since 12 = 4· 3.
√12 = √4·√3 = 2√3.
Check: 2√3≈ 3.46 and 3.462≈ 12.
Problem 8
Simplify √50-√18.
- √32
- 2√2
- 8√2
- 2√3
- 3√2
Worked answer
2√2
Simplify each term: √50 = 5√2 and √18 = 3√2.
Both now contain √2, so subtract like terms: 5√2-3√2 = 2√2.
Subtracting the insides to get √32 is a different and incorrect operation.