M
Math Coach Amy
Amy Ferguson Moncure

Algebra 1 · Unit 6

Evaluate Expressions

Evaluate expressions with absolute value, square roots, and cube roots. You do not need to rationalize a denominator.

Jump to section:

Key Ideas Easy Medium Hard Super Challenge
What to Remember

Replace, then simplify

Substitute the given number for each letter. Use parentheses, especially with a negative.

Absolute value

|x| is the distance from 0. It is never negative: |−5| = 5.

Square root and cube root

√a is the nonnegative number whose square is a. ∛a is the number whose cube is a, and it can be negative.

Leave a leftover radical

You do not need to rationalize a denominator. Simplify perfect squares and cubes under a radical when you can.

Level 1

Easy

One substitution. Integer results.

Example
Evaluate 3x + 1 when x = 4.

3(4) + 1

12 + 1 = 13

Answer: 13

1
Evaluate |x − 5| when x = 2.
Show Answer

|2 − 5| = |−3| = 3

Answer: 3

2
Evaluate √x when x = 36.
Show Answer

√36 = 6

Answer: 6

3
Evaluate ∛x when x = 8.
Show Answer

∛8 = 2

Answer: 2

4
Evaluate 2x² when x = −3.
Show Answer

2(−3)²

2(9) = 18

Answer: 18

5
Evaluate ∛x when x = −27.
Show Answer

∛−27 = −3

Answer: −3

Level 2

Medium

Two operations, or a radical that still needs a leftover.

Example
Evaluate |2x + 1| + √x when x = 16.

|2(16) + 1| + √16

|33| + 4

33 + 4 = 37

Answer: 37

1
Evaluate √x5 when x = 50.
Show Answer

√505

√50 = 5√2

5√25

Leave the 5 in the denominator. You do not need to rationalize.

Answer: 5√25

2
Evaluate x² − |x| when x = −4.
Show Answer

(−4)² − |−4|

16 − 4 = 12

Answer: 12

3
Evaluate ∛8x when x = 8.
Show Answer

∛64 = 4

Answer: 4

4
Evaluate √9x when x = 4.
Show Answer

√36 = 6

Answer: 6

5
Evaluate 12√x when x = 9.
Show Answer

12√9

123 = 4

Answer: 4

Level 3

Hard

Nested roots, two variables, or a leftover radical in the answer.

Example
Evaluate √2a + ∛b when a = 8 and b = −8.

√16 + ∛−8

4 + (−2) = 2

Answer: 2

1
Evaluate |x² − 10| when x = 3.
Show Answer

|9 − 10| = |−1| = 1

Answer: 1

2
Evaluate √183 when the expression is already numeric. Simplify the radical, but do not rationalize.
Show Answer

√18 = 3√2

3√23

Answer: 3√23

3
Evaluate 2|x − 4| − ∛x when x = 8.
Show Answer

2|8 − 4| − ∛8

2(4) − 2 = 6

Answer: 6

4
Evaluate √a² when a = −5. Why is the answer 5, not −5?
Show Answer

√25 = 5

The principal square root is nonnegative. √a² = |a|.

Answer: 5

5
Evaluate ∛x2 when x = 54.
Show Answer

∛54 = 3∛2 because 54 = 27 × 2.

3∛22

Answer: 3∛22

Super Challenge

Super Challenge

1
For which x is √x − 3 defined as a real number? Evaluate at x = 3 and at x = 12.
Show Answer

Need x − 3 ≥ 0, so x ≥ 3.

At x = 3: √0 = 0

At x = 12: √9 = 3

Answer: x ≥ 3; 0 and 3

2
A student evaluates √82 as √4 = 2. Correct the work.
Show Answer

√8 = 2√2

2√22 = √2, not 2.

You cannot cancel the 2 under the radical with the 2 in the denominator unless it is a factor outside the radical.

Answer: √2

3
Evaluate |∛x − 2| when x = −8 and when x = 8.
Show Answer

x = −8: |−2 − 2| = |−4| = 4

x = 8: |2 − 2| = 0

Answer: 4 and 0

Need help evaluating expressions?

Book a Session with Amy