Name: ______________________________
Algebra 1
Worksheet – Simplifying Radical Expressions
May 15, 2013
Radical Expressions
A radical expression is any algebraic expression that contains a radical. We indicate radicals
with the √ notation, also known as a “square root” 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. Sometimes that value will be a rational
number, as in √100 or
. 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. 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 = √2 ∙ 7
Next we simplify by taking out of the radical all perfect squares: √2 ∙ 7 = 2 ∙ 7
Finally, we evaluate this numerical expression:
2 ∙ 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
=
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 = √2 ∙ 3 = 2√3
►√162 = √2 ∙ 3 = 3 √2 = 9√2
The same process is followed for radicals containing algebraic expressions, e.g.
or
25( − 2) .
►
=
√
► 25( − 2) =
5 ( − 2) = 5( − 2)√ − 2
Here are the steps to follow in simplifying radical expressions:
1. Factor what’s inside the radical completely, including any
numerical parts and 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. Check: Only single factors should be left inside the radical.
Classroom practice problems:
1. √72
2. √560
3. √450
4. √6048
5. √
6.
7. √18
8. √32
8
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 removing all perfect squares:
15. √88
16. √700
17. √392
18. √54
19. √23
20. √576
21. √4563
22. √3960
23. √
24.
25. √2
26. √9
27. √24
28.
98
29.
50
30. √68
31.
( + 7)
32.
33.
( + )
34. √4
−2 +1
− 8 − 12