Math Lesson 7.1.3 - Negative Indices. The Meaning of Reciprocal

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Welcome to our Math lesson on Negative Indices. The Meaning of Reciprocal, this is the third lesson of our suite of math lessons covering the topic of Indices, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Negative Indices. The Meaning of Reciprocal

We have briefly mentioned in the previous tutorial the concept of negative powers when dealing with powers of algebraic fractions. In that case, we explained that if an algebraic fraction is raised at a negative power, the fraction is inverted upside down. We call the new number obtained (whether it may be a fraction or integer) the reciprocal of the original number.

By definition, the reciprocal represents a number, expression, or function so related to another that their product is unity (one, therefore). In other words, the reciprocal of a quantity represents the quantity obtained by dividing the number one by a given quantity.

For example the reciprocal of 3 is 1/3, the reciprocal of 2x is 1/2x, etc.

There is a property involving negative indices that says:

If a number or expression is raised to a negative power, it is equal to the reciprocal raised at the corresponding positive power.

In symbols,

In fractions, the reciprocal is obtained by swapping the positions of numerator and denominator. For example, the reciprocal of a/b is b/a, the reciprocal of 3/5 is 5/3 and so on.

Example 3

Calculate the value of the following expressions.

1. (3² ∙ 4³/2⁵ )^-4
2. (2⁴/5² )^-3

Solution 3

1. We have
   (3² ∙ 4³/2⁵ )^-4 = (2⁵/3² ∙ 4³)⁴
   We can express 4³ as (2²)³ = 2^{2 × 3} = 26. Thus, we obtain
   (2⁵/3² ∙ 4³)⁴ = ( 2⁵/3² ∙ 2⁶)⁴
   = (2^{5 - 6}/3² )⁴
   = (2^-1/3² )⁴
   = (1/2¹ ∙ 3²)⁴
   = (1/2 ∙ 9)⁴
   = (1/18)⁴
   = 1⁴/18⁴
   = 1/104,976
2. We have
   (2⁴/5² )^-3 = (5²/2⁴)³
   = 5^{2 × 3}/2^{4 × 3}
   = 5⁶/2¹²
   = 15,625/4,096

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