NEGATIVE EXPONENTS -
EVALUATING BY APPLYING
LAWS OF EXPONENTS

Please study
RATIONAL EXPONENTS BEFORE NEGATIVE EXPONENTS
if you have not already done so.

It is a prerequisite here.

There, we provided the explanation
for Rational Exponents.

We discussed how we can apply the same
7 Laws and the 2 Rules given for
whole number Exponents can be applied
for Fractional Exponents.

Here, we apply the Laws to evaluate numbers (expressions) with Negative Exponents.

Solved Example 1 of Negative Exponents

Evaluate : (27)^4⁄3 + (32)^0.8 + (0.8)^-1 + (0.8)⁰

Solution to Example 1 :

Let A = (27)^4⁄3 + (32)^0.8 + (0.8)^-1 + (0.8)⁰
We know
27 = 3 x 3 x 3 = 3³; 32 = 2 x 2 x 2 x 2 x 2 = 2⁵;
(0.8)^-1 = 1⁄(0.8) = 10⁄8 = 5⁄4; (0.8)⁰ = 1.
∴ A = (3³)^4⁄3 + (2⁵)^0.8 + 5⁄4 + 1.
= (3^{3 x 4⁄3}) + (2^{5 x 0.8}) + 5⁄4 + 1.
= (3⁴) + (2⁴) + 1.25 + 1.
= (3 x 3 x 3 x 3) + (2 x 2 x 2 x 2) + 2.25
= 81 + 16 + 2.25 = 99.25. Ans.

Solved Example 2 of Negative Exponents

Evaluate : {(27)^-3⁄9^-3}^1⁄3

Solution to Example 2 :

{(27)^-3⁄9^-3}^1⁄3

Solution to 5(ii) of Rational Exponents:

Let A = {(27)^-3⁄9^-3}^1⁄3
We know
a^m⁄b^m = (a⁄b)^m
∴ A = {(27⁄9)^-3}^1⁄3
We know
(a^m)ⁿ = a^mn
∴ A = (27⁄9)^{-3 x 1⁄3}
= (27⁄9)^-1 = (3)^-1
= 1⁄3. Ans.

Solved Example 3 of Negative Exponents

Evaluate : (√32 - √5)^1⁄3.(√32 + √5)^1⁄3

Solution to Example 3 :

Let A = (√32 - √5)^1⁄3.(√32 + √5)^1⁄3
We Know
a^m x b^m = (ab)^m
∴ A = {(√32 - √5)(√32 + √5)}^1⁄3
We Know
(a - b)(a + b) = a² - b²
∴ A = {(√32)² - (√5)²}^1⁄3
We Know
(√a)² = a; ∴ A = {32 - 5}^1⁄3 = (27)^1⁄3
We know 27 = 3 x 3 x 3 = 3³
∴ A = (3³)^1⁄3 = (3^{3 x 1⁄3}) [since (a^m)ⁿ = a^mn]
= 3¹ = 3. Ans.

Solved Example 4 of Negative Exponents

Evaluate : 9^5⁄2 - 3.4⁰ - (1⁄81)^-1⁄2

Solution to Example 4 of Negative Exponents:

Let A = 9^5⁄2 - 3.(4)⁰ - (1⁄81)^-1⁄2
We know
9 = 3 x 3 = 3²; (4)⁰ = 1; (1⁄81) = {1⁄(9 x 9)} = {1⁄(9)² } = (9)^-2;
∴ A = (3²)^5⁄2 - 3.(1) - (9^-2)^-1⁄2
= 3^{2 x 5⁄2} - 3 - (9^{-2 x -1⁄2}) [since (a^m)ⁿ = a^mn]
= 3⁵ - 3 - (9¹) = 3 x 3 x 3 x 3 x 3 - 3 - 9 = 243 - 12 = 231. Ans.

Exercise on Negative Exponents

1. (0.01)^-1⁄2
2. (64⁄25)^-3⁄2
3. If (9ⁿ x 3² x 3ⁿ - 27ⁿ)⁄(3^3m x 2³) = 3^-3, Prove that (m - n) = 1.

For Answers, see at the bottom of the page.

Answers to Exercise on Negative Exponents

1. 10.
2. 125⁄512.