Exponent Rules
- Introduction
- What Are the Different Exponent Rules?
- Zero Power Rule
- Product of Powers Rule
- Quotient of Powers Rule
- Power of a Product Rule
- Power of a Quotient Rule
- Power of a Power Rule
- Negative Exponent Rule
- Practice Problems
- Frequently Asked Questions
Introduction
Exponent rules are also called as exponent laws. These are fundamental laws used to simplify expressions involving exponents. These rules enable us to carry out arithmetic operations like addition, subtraction, multiplication, and division more efficiently when dealing with expressions containing exponents. By applying these rules, we can simplify complex expressions with fractional, decimal, and root exponents, making mathematical computations quicker and easier. Let's understand the various rules of exponents.
What Are the Different Exponent Rules?
There are some rules that help in solving problems related to exponents.
Zero Power Rule
If a non-zero number is raised to `0`, its value equals `1`.
Therefore, \(a^0 = 1\) for any non-zero number \(a\).
Example: Evaluate `7^0`
Explanation: Apply the zero power rule, `a^0=1`
So, `7^0=1`
Product of Powers Rule
If two numbers having the same base are multiplied, then we add the exponents of the two numbers.
Consider two powers with the same base \(a\), represented as \(a^m\) and \(a^n\), where \(m\) and \(n\) are exponents. When we multiply \(a^m\) by \(a^n\), it's like multiplying \(a\) to itself \(m\) times and then multiplying \(a\) to itself \(n\) more times:
\[a^m \times a^n = \overbrace{a \times a \times \ldots \times a}^{m \text{ times}} \times \overbrace{a \times a \times \ldots \times a}^{n \text{ times}}\]
Since we're combining the factors, we end up with \(a\) multiplied by itself a total of \(m + n\) times:
\[a^m \times a^n = \underbrace{a \times a \times \ldots \times a}_{(m + n) \text{ times}} = a^{m + n}\]
Example: Simplify `2^5\cdot 2^3`
Solution: Use the product of powers rule, `a^m\cdot a^n=a^{m+n}`
So, `2^5\cdot2^3=2^{5+3}=2^8`
Quotient of Powers Rule
If two numbers having the same base are divided, then we subtract the exponents of the two numbers.
Let's consider two powers with the same base \(a\), represented as \(a^m\) and \(a^n\), where \(m\) and \(n\) are exponents.
When we divide \(a^m\) by \(a^n\), it's akin to canceling out \(a\) from each \(a^m\) \(n\) times:
\[a^m ÷ a^n = \frac{\overbrace{a \times a \times \ldots \times a}^{m \text{ times}}} {\underbrace{a \times a \times \ldots \times a}_{n \text{ times}}}\]
As each \(a\) in the numerator cancels out with an \(a\) in the denominator, we're left with \(a\) multiplied by itself \(m - n\) times:
\[a^m ÷ a^n = \underbrace{a \times a \times \ldots \times a}_{(m - n) \text{ times}} = a^{m - n}\]
Example: Simplify `\frac{5^7}{5^3}`
Solution: Apply the quotient of power rule, `\frac{a^m}{a^n}=a^{m-n}`
So, `\frac{5^7}{5^3}=5^{7-3}=5^4`
Power of a Product Rule
When raising a product to a power, we can distribute the exponent over the different factors.
When we have a product \(ab\) raised to a power \(n\), represented as \((ab)^n\), it means we're multiplying \(ab\) by itself \(n\) times.
`\begin{align*}
(ab)^n &= \underbrace{ab \times ab \times \ldots \times ab}_{n \text{ times}} \\
&= \underbrace{a \times a \times \ldots \times a}_{n \text{ times}} \times \underbrace{b \times b \times \ldots \times b}_{n \text{ times}} \\
&= a^n \times b^n
\end{align*}`
Example: Simplify `\left(2\cdot6\right)^3`
Solution: Power of a product rule, `\left(a\cdot b\right)^n=a^n\cdot b^n`
`\left(2\cdot6\right)^3=2^3\cdot6^3`
Power of a Quotient Rule
When raising a quotient to a power, we can distribute the exponent over the numerator and the denominator.
When we raise this fraction to a power \(n\), it means multiplying \(\frac{a}{b}\) by itself \(n\) times:
`\begin{align*}
\left(\frac{a}{b}\right)^n &= \underbrace{\frac{a}{b} \times \frac{a}{b} \times \ldots \times \frac{a}{b}}_{n \text{ times}} \\
&= \frac{\underbrace{a \times a \times \ldots \times a}_{n \text{ times}}}{\underbrace{b \times b \times \ldots \times b}_{n \text{ times}}} \\
&= \frac{a^n}{b^n}
\end{align*}`
Example: Simplify `\left(\frac{6}{5}\right)^7`
Solution: Use power of a quotient rule, `\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}`
So, `\left(\frac{6}{5}\right)^7=\frac{6^7}{5^7}`
Power of a Power Rule
If a number with an exponent is raised to another exponent, then we multiply the exponents.
Think of a number \(x\) raised to a power \(m\). It means multiplying \(x\) by itself \(m\) times. Now, if we raise that expression \(x^m\) to another power \(n\), it's like multiplying \(x^m\) by itself \(n\) times. Since each \(x^m\) means \(x\) multiplied by itself \(m\) times, doing it \(n\) times is like multiplying \(x\) by itself \(m \times n\) times altogether.
\[(x^m)^n = \underbrace{x^m \times x^m \times \ldots \times x^m}_{n \text{ times}}=x^{m\times n}\]
Example: Simplify `\left(3^4\right)^2`
Solution: Apply the power of a power rule, `\left(a^m\right)^n=a^{m\cdot n}`
So, `\left(3^4\right)^2=3^{4\cdot2}=3^8`
Negative Exponent Rule
If a number has a negative number as an exponent, the exponent can be converted into a positive number by taking its reciprocal.
Imagine \(x\) as a fraction with a numerator of `1`. When raised to the power of \(n\), it's like multiplying the numerator (which is `1`) by itself \(n\) times. So, \(x^{-n}\) becomes \(\frac{1}{x^n}\), where \(x\) is the base and \(n\) is the exponent.
So, \(x^{-n}=\frac{1}{x^n}\)
Example: Write `2^{-5}` using positive exponent.
Solution: Apply the negative exponent rule, `a^{-m}=\frac{1}{a^m}`
So, `2^{-5}=\frac{1}{2^5}`
Practice Problems
Q`1`: Simplify the expression \(3^4 \times 3^2\).
- `6561`
- `729`
- `2187`
- `72`
Answer: b
Q`2`: Simplify \(5^7 \div 5^3\).
- `625`
- `25`
- `7/3`
- `125`
Answer: a
Q`3`: Simplify the expression \((2^3)^4\).
- `128`
- `256`
- `24`
- `4096`
Answer: d
Q`4`: Determine the value of \((4 \times 5)^2\).
- `40`
- `400`
- `2048`
- `100`
Answer: b
Frequently Asked Questions
Q`1`. Why are the laws of exponents important?
Answer: Understanding the laws of exponents is like having a superpower in math! These rules make it easier for us to solve problems involving numbers with powers. They're important because they help us simplify math problems and find solutions faster.
Q`2`. Are there any special cases involving exponents?
Answer: Yes, when we raise any nonzero number to the power of zero, it always equals `1`. And if we raise any number to the power of one, it stays the same.
Q`3`. How can I practice applying the exponent laws?
Answer: You can practice by solving lots of different problems involving exponents. Try simplifying expressions, figuring out the values of different powers, and solving equations with exponents.