In Mathematics, an exponent defines the number of times a number is multiplied by itself. For example, 3^{2}. It means that the number 3 has to be multiplied twice. Here, the number 3 is a base number and 2 is an exponent. The exponent can be positive or negative. In this article, we are going to discuss “Negative Exponents” in detail with its definition, rules, and how to solve the negative exponent with many solved examples.
Table of Contents:
- Negative Exponents Definition
- Rules
- Fractions with Negative Exponents
- Multiplying Negative Exponents
- How to Solve Negative Exponents?
- Practice Problems
- FAQs
Negative Exponents Definition
We know that the positive exponent tells us how many times a number is multiplied by itself. Whereas, the negative exponent tells us how many times we have to divide the base number. In other words, the negative exponent describes how many times we have to multiply the reciprocal of the base. An example of negative exponents is 3^{-2}.
Thus, 3^{-2 } is written as (1/3^{2})
Hence, the value of 3^{-2 } is 1/9.
More examples of Negative exponents:
- 5^{-1} is equal to ⅕
- X^{-4} is written as 1/x^{4}
- (2x+3y)^{-2} is equal to 1/(2x+3y)^{2}.
Negative Exponent Rules
To easily simplify the negative exponents, we have a set of rules of negative exponents to solve the problems. The following are the rules of negative exponents.
Negative Exponent Rule 1:
For every number “a” with negative exponents “-n” (i.e.) a^{-n}, take the reciprocal of the base number and multiply the value according to the value of the exponent number.
For example, 4^{-3}
Here, the base number is 4 and the exponent is -3.
According to this rule, 4^{-3} is written as 1/4^{3} = (¼)×(¼)×(¼) = 1/64
Hence, the value of 4^{-3} is 1/64.
Negative Exponent Rule 2:
For every number “a” in the denominator with negative exponent “-n” (i.e.,) 1/a^{-n}, the result can be written in the form of a×a×.. n times.
For example, 1/2^{-3}
In this example, the negative exponent is in the denominator of the fraction.
Thus, 1/2^{-3} is written as 2^{3} which is equal to 2× 2 ×2
Hence, 1/2^{-3} is equal to 8.
Fractions with Negative Exponents
The fractions with negative exponents are solved using the two above-mentioned rules. Now, let us consider the example, x^{2}/x^{-3}
Here, the numerator has a positive exponent and the denominator has a negative exponent.
By using rule 2 of negative exponents, the denominator can be written as
x^{2}/x^{-3 }= x^{2}. X^{3}
By using the laws of exponents, we can add the exponents if the base values are the same.
Thus, x^{2}. x^{3} = x^{(2+3)}
Hence, x^{2}/x^{-3 }= x^{5}
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How to Multiply Negative Exponents?
The multiplication of two negative exponents is the same as the multiplication of the two numbers. As we know that the negative exponents can be converted into fractions, it is easy to simplify the negative exponents. For multiplying the negative exponents, the first step is to convert the negative exponents to their fractional form and proceed with the process of multiplying two fractional numbers.
Now, let us consider an example.
The multiplication of two numbers with negative exponents, such as (⅔)^{-2} and (4/2)^{-3 }is as follows:
First, convert the negative exponents to positive exponents by taking the reciprocal of the given number.
Thus, (⅔)^{-2 } becomes (3/2)^{2} and (4/2)^{-3} becomes (2/4)^{3}
Now, multiply the numbers, and we get
= (3/2)^{2}× (2/4)^{3}
= (9/4)×(8/64)
Now, simplify the expression
= (9/4)×(1/8)
= 9/32.
Thus, the multiplication of two numbers with negative exponents, (⅔)^{-2} and (4/2)^{-3} is 9/32.
Note:
- The relationship between positive exponents and negative exponents is expressed as a^{n} = 1/a^{-n.}
- If a^{-n} = a^{-m}, then we can say n=m.
How to Solve Negative Exponents?
Now, let us discuss a few examples of solving the negative exponents.
Example 1:
Simplify 4x^{-2}.
Solution:
Given expression 4x^{-2}
Using the rule, a^{-n} = 1/a^{n}
4x^{-2} = 4 (1/x^{2})
4x^{-2} = 4/x^{2}.
Example 2:
Simplify the expression: (-3x^{-1} y^{2})^{2}
Solution:
Given expression: (-3x^{-1} y^{2})^{2}
Using the negative exponent rule, the expression can be written as:
(-3x^{-1} y^{2})^{2} = [(-3y^{2})^{2}]/x^{2}
(-3x^{-1} y^{2})^{2 }= 9y^{4}/x^{2}
Thus, the simplified expression is 9y^{4}/x^{2}.
Practice Problems on Negative Exponents
Solve the following problems on negative exponents:
- 2^{-2} + 3^{-2}
- (5/3)(3^{-4})
- (a^{-2}/b^{-3})^{-2}
- 4x^{2} / 2^{-2}
- 4^{-2} – 2^{3}
Frequently Asked Questions on Negative Exponents
What is meant by negative exponents?
Negative exponents tell us how many times we have to multiply the reciprocal of the base number. For example, 2^{-2}. The equivalent expression of 2^{-2} is (½)× (½).
What is the difference between the positive and negative exponents?
A positive exponent defines how many times we have to multiply the base number, whereas a negative exponent defines how many times we have to divide the base number.
Mention the rules for negative exponents?
The two rules of negative exponents are:
a^{-n} = 1/a^{n} = (1/a)×(1/a) ×… n times
1/a^{-n} = a^{n} = a × a × … n times
What is the equivalent expression of 6x^{-1}?
The equivalent expression of 6x^{-1} is 6/x.
How to represent 10 to the negative exponent of 2?
10 to the negative exponent of 2 is represented as 10^{-2}, which is equal to 1/10^{2}.
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