About Zero & Negative Exponents
What are Exponents?
The exponent of a number indicates how many times it has been multiplied by itself.
For example:
\(3^4=3\times 3\times 3\times 3\) denotes a four-fold multiplication of 3.
Similarly \(3^9=3\times 3\times 3\times 3\times 3\times 3\times 3\times 3\times 3\) denotes a nine-fold multiplication of 3.
Therefore exponents help us to denote this tedious multiplication in an easier way. The exponent is another name for the power of a number. It could be a whole number, a fraction, a negative number, or a decimal.
We can extend the idea of exponents to include zero and negative exponents as well.
Definition of Negative Exponents
The positive exponent indicates how many times a number has been multiplied by itself.
The negative exponent, on the other hand, tells us how many times we must divide the base number. In other words, the negative exponent indicates how many times the reciprocal of the base must be multiplied.
The reciprocal of \(a^{-n}\) is \(\frac{1}{a^n}\), for any integer n and any nonzero number a.
As a result, \(3^{-2}\) is written as:
\(\frac{3^{-2}\times 3^2}{3^2}=\frac{3^{-2+2}}{3^2}\) Product of powers property
\(=\frac{3^{0}}{3^2}\) Simplify
\(=\frac{1}{3^2}\) Definition of a zero exponents
\(=\frac{1}{3\times 3}=\frac{1}{9}\)
As a result, \(3^{-2}\) has a value of \(\frac{1}{9}\).
What are Zero Exponents?
There is no need to take the base value into account when simplifying an expression with a 0 as the exponent.
Any no-zero number raised to the power 0 equals 1.
\(x^0=1\) , where \(x \neq 0\) and \(x\) can be any number 1, 2, 3 and so on.
What is Negative Exponent Rule?
According to this rule, if the exponent is negative, we can make it positive by writing the same value in the denominator while the numerator is written as 1.
The rule for negative exponents is as follows:
For any non-zero number a and an integer m,
\(a^{-m}=\frac{1}{a^m}\)
Solved Negative & Zero Exponents Examples
Example 1:
Evaluate the expression \((9)^{-5}.(9)^5\)
Solution:
\((9)^{-5}.(9)^5=(9)^{-5+5}\) Product of powers property
\(=(9)^0\) Simplify
= 1 By definition of a zero exponent
Example 2:
Calculate the value of \(3^{-2}\)
Solution:
The exponent is negative in this case (i.e., -2)
As a result, \(3^{-2}\) can be written as \(\frac{1}{3^2}\).
\(3^{-2}=\frac{1}{3^2}\) by definition of a negative exponent
\(3^{-2}=\frac{1}{9}\) evaluate the power
In other words, we can say that \(a^{-n}\) is the reciprocal of \(\frac{1}{a^n}\) if “a” is a nonzero number or a nonzero rational number.
Example 3:
Calculate the value of \(5^{-3}\)
Solution:
The exponent is negative in this case (i.e., -3)
As a result,\(5^{-3}\) can also be written as \(\frac{1}{5^3}\).
\(5^{-3}=\frac{1}{5^3}\) by definition of a negative exponent
\(5^{-3}=\frac{1}{125}\) evaluate the power
Example 4:
Evaluate the given expression \(\frac{3^5}{3^7}\)
Solution:
\(\frac{3^5}{3^7}=3^{5-7}\) quotient of powers property
\(=3^{-2}\) simplify
\(=\frac{1}{3^2}\) by definition of a negative exponent
\(=\frac{1}{9}\) evaluate the power
Example 5:
Every second, a drop of water leaks from a faucet. In two hours, how many liters of water leak from the faucet?
Drop of water: \(60^{-2}\) liters
Solution:
Converting 1 hour to seconds is needed here because the unit for the water leaking is provided per second.
\(2~h\times \frac{60~m}{1~h}\times \frac{60~\text{sec}}{1~m}=7200~\text{sec}\)
7200 seconds multiplied by the rate of water leakage
\(=7200~\text{sec}.\frac{60^{-2}~L}{1~\text{sec}}\)
\(=7200.\frac{1}{60^2}~L\) by definition of a negative exponent
\(=7200.\frac{1}{3600}~L\) evaluate the power
\(=\frac{7200}{3600}~L\) multiply
= 2 simplify
In 2 hours, 2 liters of water have leaked from the faucet.
Negative exponents indicate how many times the reciprocal of the base number must be multiplied.\(2^{-2}\) as an example indicates \(\frac{1}{2}\times \frac{1}{2}\),which is the equivalent expression of \(2^{-2}\) or \(\frac{1}{4}\).
According to the zero power rule if the exponent is zero, the result is 1, regardless of the base value. For example,\(7^0=1\).
A positive exponent indicates how many times the base number must be multiplied, while a negative exponent indicates how many times the base number must be divided.