Rewriting -9^-1 Without an Exponent: A Comprehensive Guide

What is a Negative Exponent?

A negative exponent represents a reciprocal with a positive exponent. For example, -9^-1 means 1 divided by 9 raised to the power of 1. This can be written mathematically as:

-9^-1 -1/9^1 -1/9.

Let's break this down step by step to better understand the concept.

Understanding the Division

When dealing with a negative exponent, such as -9^-1, you are essentially taking the reciprocal of the number. To understand this, let's first look at a positive exponent scenario:

Positive Exponent Example:

For -8^2, you would calculate:

1 ÷ -8 ÷ -8 This simplifies to 1/64.

A quicker method involves directly calculating the positive exponent and then taking its reciprocal. For -8^2:

-8 × -8 64 The reciprocal is 1/64.

Applying this to -9^-1 will give us:

-1/9

Using the Laws of Indices

In algebra, the laws of indices state that:

Law of Indices for Negative Exponents:

a^-b 1 / a^b

where:

a is the base, b is the exponent.

Given this law, we can rewrite -9^-1 using algebraic simplification:

-9^-1 -1 / 9^1 -1/9.

Conclusion

In conclusion, rewriting -9^-1 without an exponent involves recognizing the negative exponent as a reciprocal. By calculating the positive exponent and then taking its reciprocal, we find:

-9^-1 -1/9.

This method is not only effective but also allows for a deeper understanding of exponents and their applications in algebra.