1-5 1 Exponent Rules Explained
Key Concepts of Exponent Rules
Exponent rules are fundamental principles that govern the behavior of exponents in mathematical expressions. Understanding these rules is crucial for simplifying and solving complex algebraic expressions.
1. Product of Powers Rule
When multiplying two expressions with the same base, add the exponents:
\[ a^m \times a^n = a^{m+n} \]
Example:
\[ 3^2 \times 3^4 = 3^{2+4} = 3^6 \]
2. Quotient of Powers Rule
When dividing two expressions with the same base, subtract the exponents:
\[ \frac{a^m}{a^n} = a^{m-n} \]
Example:
\[ \frac{5^6}{5^2} = 5^{6-2} = 5^4 \]
3. Power of a Power Rule
When raising a power to another power, multiply the exponents:
\[ (a^m)^n = a^{m \times n} \]
Example:
\[ (2^3)^2 = 2^{3 \times 2} = 2^6 \]
4. Power of a Product Rule
When raising a product to a power, raise each factor to that power:
\[ (a \times b)^n = a^n \times b^n \]
Example:
\[ (4 \times 3)^2 = 4^2 \times 3^2 = 16 \times 9 = 144 \]
5. Power of a Quotient Rule
When raising a quotient to a power, raise both the numerator and the denominator to that power:
\[ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \]
Example:
\[ \left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27} \]
6. Zero Exponent Rule
Any non-zero number raised to the power of zero is 1:
\[ a^0 = 1 \]
Example:
\[ 7^0 = 1 \]
7. Negative Exponent Rule
A negative exponent indicates the reciprocal of the base raised to the positive exponent:
\[ a^{-n} = \frac{1}{a^n} \]
Example:
\[ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \]
Practical Applications
Understanding exponent rules is essential for solving real-world problems. For instance, calculating compound interest, determining the volume of a cube, or analyzing scientific data all involve exponent rules.
Example:
If you invest $1000 at an annual interest rate of 5% compounded annually for 3 years, you can calculate the future value using the formula \( A = P(1 + r)^t \):
\[ A = 1000(1 + 0.05)^3 = 1000 \times 1.05^3 \approx 1157.63 \]