Negative Exponents: Numbers & Expressions

Negative exponent is used to write the reverse of a number that has been raised to a positive exponent. In other words, when a base number is raised to a negative exponent, it means that the base should be used as the denominator of a fraction with the exponent's absolute value as the numerator. 

• The result is the reverse of the value found by raising the base to a positive exponent.
• Consider the expression 2^-3, which means "two raised to the power of negative three."
• This means that the base number, 2, is put in the denominator and raised to the power of 3 in the numerator. 
• It is the same as dividing 1 by 2³, which gives you 1/8. 

Keyterms: Negative exponent, positive exponent, number, denominator, fraction, numerator, absolute value, base number

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Negative Exponents 

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The exponent of a number represents the number of times the base must be multiplied by itself. For example, the base in the expression 8² is 8, and the exponent is 2. This is equivalent to 8*8, which equals 64.

Example: In the expression 8^-2, the base is 8, while the exponent is -2. To interpret this, the reciprocal of the base (1/8) must be multiplied by itself twice. 8^-2 can therefore be written as (1/8) (1/8), which simplifies to 1/64.

Negative Exponents

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Numbers and Expressions with Negative Exponents

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In mathematics, exponents play a crucial role in expressing repeated multiplications of a base number.

 

• A positive exponent denotes the number of times the base should be multiplied by itself
• A negative exponent introduces the concept of multiplying the base's reciprocal by a given number. Here is a table-based example.

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Rules for Negative Exponents

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Here are the rules listed below:

Rule 1

For a base 'a' with a negative exponent -n, take the reciprocal of the base (1/a) and multiply it by itself n times: a^-n = 1/a × 1/a × ... n times = 1/aⁿ.

• A number with base "a" and a negative exponent "-n," like a^-n.
• Take the reciprocal of the base as the first step. Divide 1 by a number, get its opposite. So, the reciprocal of the base "a" is 1/a.
• Now, multiply this ratio (1/a) by itself "n" times. To put it another way, you are doubling it by itself "n" times.

Example 1: Simplify the expression 5^(-3) + 6^(-3).

Solution: Apply Rule 1 for each term:

5^(-3) = 1/5³ = 1/125

6^(-3) = 1/6³ = 1/216

Adding these reciprocals: 1/125 + 1/216 = (216 + 125) / (125 * 216) = 341 / 27000.

Therefore, 5^(-3) + 6^(-3) = 341/27000.

Rule 2

The same rule applies when there is a negative exponent in the denominator: 1/a^-n = a × a × ... n times = aⁿ.

• Take a fraction in which the denominator has a negative exponent: 1/a^-n.
• Change the negative exponent to a positive one. This is the most important step. To make a-n, change it to an.
• Since the exponent is now positive, you multiply the base "a" by itself "n" times.

Example 2: Simplify the expression (1/x)^(-2) + (1/y)^(-2) .

Solution: Apply Rule 2 for each term:

(1/x)^(-2) = x²

(1/y)^(-2) = y²

Adding these results: x² + y².

Therefore, (1/x)^(-2) + (1/y)^(-2) simplifies to x² + y².

Rules for Negative Exponents

Also Read: Multiplication and Division of Integers

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Fractions with Negative Exponents

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A negative exponent gives the number's reciprocal. In other terms, a^-n equals 1/aⁿ, while 5^-3 becomes 1/5³ = 1/125. Negative exponents transform integers into fractions in this manner.

• Start with a number like a^-n.
• The negative exponent tells one to think about the reciprocal of "a", which is 1/a. 
• When a fraction is reciprocal, it is turned upside down. In other words, if it is 2/3 then it would be its opposite 3/2 .
• Now, change "a^-n" with "1/aⁿ," which is the reciprocal of "a^-n." This is an exponent that goes up!

Example: Express 3^-2 and 10^-3 as fractions.

Solution: To express 3^-2 as a fraction, we will write it as 1/3². Similarly, to express 10^-3 as a fraction, we write it as 1/10³.

Therefore, negative exponents change the numbers to fractions by taking the reciprocal of the number with the positive exponent.

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Multiplying Negative Exponents

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Negative exponents are multiplied in the same manner as other numbers. Negative exponents can be expressed as fractions, it is straightforward to solve them after fractionalization. 

• Following this conversion, negative exponents are multiplied using the same multiplication rule as positive exponents. 
• Let's examine the multiplication of negative exponents using the example below.

Example: Solve: (-2/3)^-4 × (3/4)^-2

Solution: First, convert the negative exponents to positive exponents by writing the expression in its reciprocal form: (3/4)⁴ × (4/3)²

(3/4) × (3/4) × (3/4) × (3/4) × (4/3) × (4/3)

(3 × 3 × 3 × 3) / (4 × 4 × 4 × 4) × (4 × 4) / (3 × 3)

(81 / 256) × (16 / 9)

(81 × 16) / (256 × 9)

1296 / 2304

9 / 16

Therefore, (-2/3)^-4 × (3/4)^-2 is equal to 9/16.

Also Read: Uses of Exponents to Express Small Numbers in Standard Form

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How to Solve Negative Exponents?

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To simplify expressions containing negative exponents, you must first convert them to positive exponents using one of the following rules.

• a^-n = 1/aⁿ
• 1/a^-n = aⁿ

Example: Solve: (2^-3/5) × (5^-2)

Solution: First, we simplify and turn negative exponents to positive.

Given: (2^-3/5) x (5^-2). Use the rule to turn negative exponents positive: a = 1/a (1/2³/5) × (1/5²)

Simplify exponents:

(1/2^(3/5)) × (1/5²)

Multiply fractions:

1/(2^(3/5) × 5²) 

Assessing exponents:

1/(2^(3/5) × 25)

Simplify by rewriting 

2^(3/5) as the fifth root of 2³: 1/((∛2)³ × 25)

Simplify: 

2³ cube root: 1/(2 × 25) = 1/50

So the final result is 1/50.

Also Read:

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Things to Remember

• A negative exponent indicates taking the reciprocal of the base raised to the positive exponent.
• To convert a negative exponent to a positive exponent, use the rule a^(-n) = 1/aⁿ.
• When multiplying numbers with negative exponents, convert them to positive exponents first.
• The product of numbers with negative exponents is calculated using the same rules as positive exponents.
• Negative exponents can be expressed as fractions, where the numerator is 1 and the denominator is the base raised to the positive exponent.
• Simplify expressions with negative exponents by converting them to positive exponents 
• Apply the appropriate mathematical operations.

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