Basic Mathematics & Logarithm
Indices and Surds
Grade 11
Question:
<p>The numerical value of \(a^{\frac{1}{b-c}} \times a^{\frac{1}{c-a}} \times a^{\frac{1}{a-b}}\) is (\(a, b, c\) are distinct real numbers)</p>
Step-by-Step Solution
Key Concept: Add the exponents. Show that 1/(b-c) + 1/(c-a) + 1/(a-b) = 0, so a^0 = 1.
Notice that the best first move is to reveal the hidden structure in the expression. A clever move here is to rewrite the problem in the form where the standard theorem or identity applies cleanly. Sum of exponents = $\dfrac{1}{b-c}+\dfrac{1}{c-a}+\dfrac{1}{a-b}$. Common denominator = $(b-c)(c-a)(a-b)$. Numerator = $(c-a)(a-b)+(a-b)(b-c)+(b-c)(c-a)$. Expanding and simplifying gives 0. Hence the product = $a^0 = 1$. Now, we invoke the power of that idea, simplify patiently, and then check that the final answer really fits the original problem.
Correct Answer: 1