Negative indices
Negative Indices
Understanding the Concept
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A number with a negative index represents the reciprocal of the number raised to that index when it’s positive.
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This is a powerful concept in algebra and is used extensively in solving algebraic problems.
Working with Negative Indices
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If a number ‘a’ has a negative index ‘n’, it is equivalent to 1 divided by (a^n), where ‘n’ is positive. In other words, a^-n equals 1/a^n.
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For example, 2^-3 is equivalent to 1/2^3. Therefore 2^-3 equals 1/8.
Rules for Negative Indices
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When multiplied, indices can be added together. For example, a^n * a^m equals a^(n+m).
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Therefore, if a ∈ R and a ≠ 0, a^m * a^-n equals a^(m-n).
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Similarly, a division law can be adapted for negative indices: a^m / a^n equals a^(m-n).
Applications of Negative Indices
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Negative indices are extremely useful in dealing with expressions involving very large or very small numbers, like in scientific notation.
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They also come into play when solving algebraic problems or simplifying algebraic expressions.
Examples
- Consider the expression 5^-2. Following the rule of negative indices, this equals 1/5^2, hence simplifies to 1/25.
Remember
- Avoid confusion between negative indices and negative bases. -3^2 is not the same as (-3)^2. The first expression equals -9, as the negative sign is not part of the base. The second expression equals 9, the square of -3.
Conclusion
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Grasping the concept of negative indices is necessary for simplifying complex algebraic equations or expressions.
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Check your work by using the definition of negative indices, converting back into a positive index to verify your solution.