NEGATIVE EXPONENTS -
EVALUATING BY APPLYING
LAWS OF EXPONENTS

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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.
= (33 x 4⁄3) + (25 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.

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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
ambm = (ab)m
∴ A = {(27⁄9)-3}1⁄3
We know
(am)n = amn
∴ A = (27⁄9)-3 x 1⁄3
= (27⁄9)-1 = (3)-1
= 1⁄3. Ans.





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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
am x bm = (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 = (33 x 1⁄3) [since (am)n = amn]
= 31 = 3. Ans.

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Solved Example 4 of Negative Exponents

Evaluate : 95⁄2 - 3.40 - (1⁄81)-1⁄2

Solution to Example 4 of Negative Exponents:

Let A = 95⁄2 - 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
= 32 x 5⁄2 - 3 - (9-2 x -1⁄2) [since (am)n = amn]
= 35 - 3 - (91) = 3 x 3 x 3 x 3 x 3 - 3 - 9 = 243 - 12 = 231. Ans.

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Exercise on Negative Exponents

  1. (0.01)-1⁄2
  2. (64⁄25)-3⁄2
  3. If (9n x 32 x 3n - 27n)⁄(33m x 23) = 3-3, Prove that (m - n) = 1.

For Answers, see at the bottom of the page.

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Answers to Exercise on Negative Exponents

  1. 10.
  2. 125⁄512.