Indices. The Power of Exponents. IntroductionIndices, or exponents, are a fundamental concept in mathematics that represent repeated multiplication. The laws of indices govern how to manipulate exponents in various operations. Understanding these laws is essential for simplifying expressions and solving equations. For example, in the expression 2^3, 2 is the base and 3 is the exponent.Basic Laws of IndicesProduct Rule: When multiplying powers with the same base, add the exponents.[latex]a^{m} times a^{n}= a^{(m+n)}[/latex].Quotient Rule: When dividing powers with the same base, subtract the exponents.[latex]a^{m} div a^{n} = a^{(m-n)}[/latex].Power of a Power Rule: When raising a power to another power, multiply the exponents.[latex](a^{m})^{n} = a^{(mtimes n)}[/latex].Zero Exponent Rule: Any non-zero number raised to the power of 0 is equal to 1.[latex]a^{0} = 1[/latex] (where [latex]a ≠ 0[/latex]).Negative Exponent Rule: A number raised to a negative exponent is equal to its reciprocal raised to the positive exponent.[latex]a^{-m} = frac{1}{a^{m}}[/latex].Solved ExamplesSimplify: [latex]2^{3} times 2^{5}[/latex].Using the product rule: [latex]2^{(3+5)} = 2^{8}[/latex].Simplify: [latex]3^{7} div 3^{4}[/latex].Using the quotient rule: [latex]3^{(7-4)} = 3^{3}[/latex].Simplify: [latex](x^2)^{3}[/latex].Using the power of a power rule: [latex]x^{(2×3)} = x^{6}[/latex].Evaluate: [latex]5^{0}[/latex].Using the zero exponent rule: [latex]5^{0} = 1[/latex].Simplify: [latex]2^{-4}[/latex].Using the negative exponent rule: [latex]frac{1}{2^{4}} = frac{1}{16}[/latex].Additional Laws of IndicesPower of a Product Rule: [latex](ab)^{m} = a^{m} times b^{m}[/latex].Power of a Quotient Rule: [latex](frac{a}{b})^{m} = a^{m} div b^{m}[/latex].ConclusionThe laws of indices provide a powerful tool for simplifying expressions and solving equations involving exponents. By understanding and applying these laws, you can effectively manipulate exponents and achieve efficient mathematical calculations.EXERCISE 15.0.Simplify: [latex]a^{3}times a^{5}[/latex]Evaluate: [latex]2^{4}times 2^{2}[/latex]Find the product: [latex]x^7 times x^{−2}[/latex]Simplify: [latex]b^{8} div b^{3}[/latex]Evaluate: [latex]5^{6} div 5^{4}[/latex]Find the quotient: [latex]y^{−5} div y^{−8}[/latex]Simplify: [latex] (x^{2})^{3}[/latex]Evaluate: [latex] (3^{4})^{2}[/latex]Find the result: [latex] (a^{−3})^{4}[/latex]Simplify: [latex]x^{0}[/latex]Evaluate: [latex]5^{0}[/latex]Find the value: [latex] (a^{3}+b^{2})^{0}[/latex]Simplify: [latex]a^{−4}[/latex]Evaluate: [latex]2^{-3} [/latex]Find the equivalent expression: [latex]x^{−5}[/latex]Simplify: [latex] (x^frac{1}{2})^{4}[/latex]Evaluate: [latex] (2^{3})^frac{1}{2}[/latex]Find the result: [latex] (a^{-frac{2}{3}})^{6}[/latex]Simplify: [latex]a^{−frac{1}{2}}[/latex]Evaluate: [latex]2^{-frac{3}{4}} [/latex]Find the equivalent expression: [latex]x^{−{2.5}}[/latex]Submit Answers via Chat e.g Exercise 15.o, then type Answers (Number your Answers).FOR MORE CONTENT!!!Register HERE Now! Pick a course, watch our Videos and take our CBT's.Additional Resources:N/A: https://www.waokmath.com[Image Source]: https://www.freepik.com/.
