Radical Functions & Equations — Practice
Squaring both sides is legal, and it can still hand you an answer that is wrong.
What these questions cover
These questions use 5 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Isolate — Get the variable alone on one side.
- Square root — For k ≥ 0, √k is the nonnegative number whose square is k.
- Radical — The √x symbol, and the expression under it.
- Roots — The x values where y = 0.
- Rational exponent — x1/2 = √x and xm/n = √[n]xm.
Problem 1
Solve √x=7.
Worked answer
49
x=49
Check: √49=7
Problem 2
Solve √x+5=4.
Worked answer
11
x+5=16
x=11
Problem 3
At what x does y=√x-3 begin?
Worked answer
3
x-3 ≥ 0
x ≥ 3
Problem 4
Evaluate 251/2.
Worked answer
5
√25=5
Problem 5
Evaluate 271/3.
Worked answer
3
√[3]27=3
Problem 6
Solve √2x-1=5.
Worked answer
13
2x-1=25
2x=26
x=13
Problem 7
Solve √x+3=8.
Worked answer
25
√x=5
x=25
Problem 8
Evaluate 82/3.
Worked answer
4
√[3]8=2
22=4
Problem 9
Evaluate 163/4.
Worked answer
8
√[4]16=2
23=8
Problem 10
Solve √x+6=x. Give the only valid answer.
Worked answer
3
x+6=x2
(x-3)(x+2)=0
x=-2 fails since a root is never negative
Problem 11
How many solutions does √x=-4 have?
- None
- One
- Two
- Infinitely many
Worked answer
None
A principal square root is never negative
Squaring gives x=16, but √16=4 ≠ -4
Problem 12
Solve √3x+1=x-1. Give the only valid answer.
Worked answer
5
3x+1=x2-2x+1
x2-5x=0 so x=0 or x=5
x=0 gives 1=-1, which fails
Problem 13
Solve √[3]x= -2.
Worked answer
-8
Cube both sides
x=-8
Problem 14
Write √[5]x3 using a rational exponent.
Worked answer
x^(3/5)
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Problem 15
Solve 2√x-1 = 9.
Worked answer
25
2√x=10
√x=5
x=25
Problem 16
For what values of x is √5-x defined? Give the largest allowed x.
Worked answer
5
5-x ≥ 0
x ≤ 5