Fractions raised to a negative index

Introduction to Fractions Raised to a Negative Index

  • Applying a negative index to fractions, essentially instructs us to take the reciprocal of the fraction.
  • The fundamental property that makes this work is that any non-zero number raised to the power of -1 is simply its reciprocal.

Understanding the Rule

  • The general rule here is that a^-n = 1/a^n where a is a non-zero real number and n is a positive integer.
  • So for fractions, (a/b)^-n = (b/a)^n.
  • This rule must be applied when simplifying expressions involving fractions raised to negative powers.

Example Case

  • Consider the expression, (3/4)^-2.
  • Applying the rule we get (4/3)^2.
  • Which simplifies to 16/9.

Further Observations

  • When simplifying equations that contain fractions raised to a negative index, it’s essential to always remember to invert the fraction before applying the power.
  • Extra concentration is needed to avoid careless errors when working with negative indices and fractions.
  • Always make sure to go through the simplification process to get your answer in the simplest form. This technique is beneficial not only in obtaining the correct answer but also in terms of understanding the problem better.

Practising Fractions Raised to a Negative Index

  • As with all areas of mathematics, mastering the handling of negative indices and fractions requires regular practice.
  • Ensure to work through an extensive range of sample problems and exam-style questions to familiarise yourself with different problem scenarios.
  • You should be comfortable applying the rule consistently and simplifying expressions involving fractions raised to negative powers. Then, working on more complex algebraic expressions involving negative indices will become significantly more manageable.

Final Notes

  • Working with negative indices and fractions may initially seem challenging, but with consistent practice and application of the rule, it becomes much easier.
  • It is essential to remember that fractions raised to a negative index should first be inverted before raising to the power.
  • Lastly, always strive for accuracy when conducting mathematical operations and simplifying your answers. The value of understanding fractions raised to a negative index extends well beyond the classroom, as it is also commonly used in more advanced mathematical and scientific computations.