Math 7 – accel.
Worksheet – radical expressions
April 4, 2012
Radical Expressions
A radical expression is any algebraic expression that contains a radical. We indicate radicals with the √
notation, also known as a radical symbol. The expression inside the radical is called the radicand – when the
radicand is a positive number, we can evaluate the radical, meaning that we can find a value for the radical.
1
Sometimes that value will be a rational number, as in √100 or √4. In other cases the value of a radical may be
an irrational number, such as √2, which can be approximated with a decimal value (√2 ≈ 1.4142135 …). In
such cases, the only way to denote the exact value of the radical is by leaving it in radical form (e.g. √2). When
the radicand is a negative number, it cannot be evaluated as a real number – when the solution to an equation
yields a radical with a negative radicand, the equation has no solution.
Examples: Evaluate the following radicals. If the value is an irrational number, use a calculator to find the
approximate value and round to the nearest thousandth. If the value is not a real number, state “not a real
number.”
►Evaluate √784.
First we factor 784 (using, for instance, a factor tree):
√784 = √24 ∙ 72
Next we simplify by taking out of the radical all perfect squares: √24 ∙ 72 = 22 ∙ 7
Finally, we evaluate this numerical expression:
22 ∙ 7 = 4 ∙ 7 = 28
►Evaluate √15.
Since 15 does not contain any perfect square factors, it cannot be evaluated to a rational number.
Using a calculator, we find that √15 ≈ 3.873.
►Evaluate √25 − 75.
Since the radicand has a negative value, the radical cannot be evaluated as a real number. The correct
response is “not a real number.”
►Note that when we take a perfect square out of a radical, its exponent is halved. This is a result of the
simple rule that states that
√𝑥 2 = 𝑥
Simplifying Radical Expressions
A radical expression that cannot be evaluated as a rational number must be written in simplified form. A
radical expression in simplified form cannot contain any factors inside the radical that are perfect squares –
any perfect squares must be “pulled out” of the radical. This is easily done for numerical radical expressions,
e.g. √12 and √162.
►√12 = √22 ∙ 3 = 2√3
►√162 = √2 ∙ 34 = 32 √2 = 9√2
The same is process is followed for radicals containing algebraic expressions, e.g. √𝑥 3 𝑦 4 or √25𝑛3 .
►√𝑥 3 𝑦 4 = 𝑥𝑦 2 √𝑥
►√25𝑛3 = √52 𝑛3 = 5𝑛√𝑛
Here are the steps to follow in simplifying radical expressions:
1. Factor the radicand completely. This includes any
numerical parts, as well as any polynomials that can be factored.
2. For any factors that appear two or more times, bring out
pairs of factors. When bringing out pairs of factors, each pair
becomes a single factor outside of the radical.
3. Only single factors may be left inside the radical.
Classroom practice problems:
1. √20
2. √72
3. √32
4. √108
5. √𝑎𝑏 2
6. √8𝑦 2
7. √18𝑥 3
8. √32𝑛4 𝑘 7
Homework problems.
A. Evaluate each radical. For irrational values, use a calculator and round to the nearest thousandth:
9. √121
10. √55
11. √−16
12. √289
13. √100 − 81
14. √25 − 36
B. Simplify these radicals by factoring and then removing all perfect squares:
15. √28
16. √40
17. √48
18. √27
19. √232
20. √484
21. √128
22. √117
23. √𝑥 2
24. √𝑦 3
25. √2𝑎2
26. √9𝑏 3
27. √24𝑐 4
28. √98𝑥𝑦 2
29. √50𝑛2 𝑝5
30. √68𝑠 3 𝑡 6
31. √(𝑥 − 5)2
32. √4𝑦 3 (𝑦 + 1)