Lesson 9 – Negative Exponents and Reciprocals
Specific Outcome:
1
, a ≠ 0 (3.1).
-
Explain, using patterns, why a-n =
-
Identify and correct errors in a simplification of an expression that involves powers (3.6).
𝑎𝑛
Powers with Negative Exponents:
When a is any non-zero number and n is a rational number, a-n is the reciprocal of an.
𝑎−𝑛 =
1
1
𝑎𝑛𝑑
= 𝑎−𝑛 , 𝑎 ≠ 0
𝑎𝑛
𝑎𝑛
Example 1: Write each power with a positive exponent.
a) 2−2
b) 5−3
1
1 −3
1
d) 4−2
c) (−1.5)−3
f) (3)
e) 2−3
Powers with Fractional Base and Negative Exponents:
𝑎 −𝑛
𝑏 𝑛
( ) = ( ) 𝑎, 𝑏 ≠ 0
𝑏
𝑎
Example 2: Write each power with a positive exponent.
3 −2
5 −3
a) (4)
b) (2)
10 −3
c) ( 3 )
𝑥 −3
d) (2)
Practice Questions: Page 233 # 3, 5, 6, 7, 8
Negative Rational Exponents:
We can apply the meaning of rational exponents and negative exponents to evaluate
powers with negative rational exponents.
𝑎
−
𝑚
𝑛
𝑚
=
1 𝑛
(𝑎)
𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡 𝑖𝑠 negative
𝑚
𝑛
1
= ( √𝑎)
𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡 𝑖𝑠 𝑎 rational 𝑛𝑢𝑚𝑏𝑒𝑟
2
We can evaluate this power in steps as follows:
E.g., 8−3
1. Write with a positive exponent:
2. Write as a radical (fractional exponent):
3. Simplify radical in brackets:
4. Evaluate:
Example 3: Evaluate the following powers.
a) 4
−
1
2
b) 49
9 −0.5
Example 4: (16)
a)
−
3
2
3
c)
16 −2
(25)
is equal to,
256
81
4
b) 3
c) −
4
3
81
d) 256
Example 5: Paleontologists use measurements from fossilized dinosaur tracks and the
5
−7
formula 𝑣 = 0.155𝑠 3 𝑓 6 to estimate the speed at which the dinosaur travelled. In the
formula, 𝑣 is the speed in metres per second, 𝑠 is the distance between successive
footprints of the same foot, and 𝑓 is the foot length in metres. Estimate the speed of the
dinosaur when 𝑠 = 1 𝑎𝑛𝑑 𝑓 = 0.25.
Practice Questions: Page 233 # 12, 13, 16, 21(a)