Laws of Exponents for Real Numbers
This section details the laws of exponents and how to apply them to real numbers. Key concepts of exponents are first reviewed, including the definitions and basic laws learned in earlier classes. The laws established include:
- Product of Powers: When multiplying two powers with the same base, add the exponents: a^m * a^n = a^{m+n}.
- Power of a Power: When raising a power to another power, multiply the exponents: (a^m)^n = a^{mn}.
- Quotient of Powers: When dividing two powers with the same base, subtract the exponents: a^m / a^n = a^{m-n}.
- Power of a Product: When raising a product to a power, distribute the exponent to both factors: (ab)^m = a^m * b^m.
The section also addresses the concept of zero and negative exponents, introducing definitions that allow for rational numbers to have fractional exponents: n√a = a^{1/n}.
Finally, examples provide guidance in simplifying expressions with rational and negative exponents. Students are encouraged to utilize these laws to solve problems involving exponents effectively.
Example 21: Simplify
(i) \( \frac{2^4 \cdot 2^3}{2^6} \)
(ii) \( \frac{3^7}{3^2 \cdot 3^3} \)
(iii) \( \frac{5^9}{(5^3)^2} \)
(iv) \( \frac{7^4 \cdot 7^5}{7^6} \)
Solution :
(i) \( \frac{2^4 \cdot 2^3}{2^6} = \frac{2^{4+3}}{2^6} = 2^{7-6} = 2^{1} = 2 \)
(ii) \( \frac{3^7}{3^2 \cdot 3^3} = \frac{3^7}{3^{2+3}} = 3^{7-5} = 3^{2} = 9 \)
(iii) \( \frac{5^9}{(5^3)^2} = \frac{5^9}{5^{3 \cdot 2}} = 5^{9-6} = 5^{3} = 125 \)
(iv) \( \frac{7^4 \cdot 7^5}{7^6} = \frac{7^{4+5}}{7^{6}} = 7^{9-6} = 7^{3} = 343 \)