Key Concepts:
Product Rule:
When multiplying powers with the same base, add the exponents: \[ a^m \cdot a^n = a^{m+n} \]Quotient Rule:
When dividing powers with the same base, subtract the exponents: \[ \frac{a^m}{a^n} = a^{m-n} \quad (a \neq 0) \]Power of a Power:
Multiply the exponents: \[ (a^m)^n = a^{mn} \]Zero Exponent:
Any non-zero base raised to the power of 0 is 1: \[ a^0 = 1 \quad (a \neq 0) \]Negative Exponent:
A negative exponent indicates a reciprocal: \[ a^{-n} = \frac{1}{a^n} \]Fractional Exponent:
A fractional exponent represents a root (find more in Chapter 1.3): \[ a^{\frac{m}{n}} = \sqrt[n]{a^m} \]Example 1: Applying the Laws of Indices
Simplify: \[ \frac{3^4 \cdot 3^{-2}}{3} \]Solution:
Step 1: Use the product rule on the numerator: \[ 3^4 \cdot 3^{-2} = 3^{4 + (-2)} = 3^2 \] Step 2: Divide by \(3 = 3^1\): \[ \frac{3^2}{3^1} = 3^{2-1} = 3^1 = 3 \] Therefore, the answer is: \[ 3 \]Example 2: Zero and Negative Exponents
Simplify: \[ 2^0 + 5^{-2} \]Solution:
Step 1: Apply the zero exponent rule: \[ 2^0 = 1 \] Step 2: Apply the negative exponent rule: \[ 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \] Step 3: Add the results: \[ 1 + \frac{1}{25} = \frac{26}{25} \]Example 3: Fractional Exponents
Simplify: \[ 16^{\frac{3}{4}} \]Solution:
Step 1: Use the fractional exponent rule: \[ 16^{\frac{3}{4}} = \left( \sqrt[4]{16} \right)^3 \] Step 2: Simplify: \[ \sqrt[4]{16} = 2 \quad \Rightarrow \quad 2^3 = 8 \] Therefore, the answer is: \[ 8 \]