Lesson 5 · from §10 of the 1910 text
Evaluating Expressions
Finding the numerical value of an expression for given values of its letters comes up all the time. Later, you'll also use it to check the results of algebraic work.
To evaluate an expression, replace each letter with its value, then simplify using the order of operations from the last lesson.
Worked example
Find the value of a^2 + 2c when a = 3 and c = 5.
a^2 + 2c = 3^2 + 2 \cdot 5 = 9 + 10 = 19
Quick check
Find the value of 4a + c when a = 2 and c = 7.
Watch the difference between exponents and coefficients: with a = 3, a^2 = 9 but 2a = 6.
Quick check
When a = 5, find a^2 and 2a.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
In Exercises 1–23, let a = 3, b = 1, c = 5, d = 7, and e = 2. Substitute each value and simplify.
4a + 3d
a^2 + 2c
ab + cd
c^2 - 5ab
abcd - 4e^2
\dfrac{d + c}{e}
\dfrac{cde}{5e} + \dfrac{ace}{2a}
\dfrac{10b}{e} - \dfrac{c}{b}
\dfrac{1}{c} + \dfrac{1}{d} + \dfrac{1}{e}
\dfrac{e^2}{ac} + \dfrac{4cd}{ae}
\dfrac{cd + ae + ce}{a - b + c}
\dfrac{a^2 + b^2 + c^2 + d^2}{a + b + c + d}
\dfrac{a^3 - ce^2 + 3cd}{d - a + c}
5b^3 + 4b^2 - 2b - 5
3b^5 - 14b^4 + 11b^3 + 11b^2 + 13b - 20
a^e
c^a
d^e + e^a
b^c + c^a
d^e - c^2 + b^3
e^2 \cdot c^a
\dfrac{d + e^c}{2c + a}
\dfrac{a^a + 3b}{c^e - d - e - b}
Find the numerical value of each expression when a = 4, b = 0, c = 5, d = 7, and e = 8.
\dfrac{4a + 3b + 2d}{c + e + 2}
\dfrac{b}{a + c + d}
abc + acd - be
\dfrac{ab}{c} + \dfrac{bd}{a} + \dfrac{be}{cd}
\dfrac{a^2 - b^2 + c^3}{3d - 2e + c}
\sqrt{a} + \sqrt{2e}
2\sqrt{a} + \sqrt[3]{e}
b\sqrt{a} + cd + \sqrt[3]{e} \cdot \sqrt{a}
c\sqrt[2]{2ae} + d\sqrt[3]{2ae}
\sqrt{b^2 + c^2 + d + a}
\sqrt[3]{3d + ec + a - 1}
de + ac\sqrt{ac^2 + 2e + c}
(a + c)c
(a + e)(c + d)
ab(a + b)
acd(a + c + d)
ac(e - c + b + d)
(a + d)^2
(e - a)^2
(d - a)^3
\dfrac{3ad(3a - 2c)^2(e - c)^3}{6ed}
\dfrac{ab(c + d)(a + b)}{e - a}
c^3 - 3c^2a + 3ca^2 - a^3
Check whether each equation is true for the given values.
If x = 2 and y = 3, does 3x + 5y = 21?
If x = 8, does 7x - 9 = 3x + 25?
Does x^2 + 5x + 6 = 0 if x = 2? If x = 3? If x = 4?
Does 3x^2 - 14x - 5 = 0 if x = 5? If x = \tfrac{1}{3}? If x = 6? If x = 4?