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To fully understand negative exponents, it helps to first grasp the concept of a reciprocal. The reciprocal of a number is essentially one divided by that number. So, if you have a number represented by the letter x, its reciprocal is written as \( \frac{1}{x} \). Understanding the reciprocal of a number becomes crucial when working with negative exponents, as they involve transforming numbers into their reciprocals with positive exponents.

Here are some simple examples to illustrate the idea:

• The reciprocal of 2 is \( \frac{1}{2} \).
• The reciprocal of 5 is \( \frac{1}{5} \).
• The reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \).

This concept will be helpful in the next sections where we explore how to deal with negative exponents.

In mathematics, when we talk about exponents, we are referring to a number written in a form like \( t^5 \). Here, t is the **base**, and 5 is the **exponent**. The base is the number that is being multiplied by itself, and the exponent tells us how many times the base is multiplied. So, in \( t^5 \), t is multiplied by itself 5 times:

\[ t^5 = t \times t \times t \times t \times t \].

Now, in our given problem:\( \frac{1}{t^5} \), t is still the base, and 5 is still the exponent, but since it is under 1, it implies we will use a negative exponent to shift it back to the numerator.

The negative exponent rule is a fundamental concept to convert expressions with negative exponents to positive exponents and vice versa. It states: For any non-zero number x and positive integer n, \( x^{-n} = \frac{1}{x^n} \).

This rule helps us rewrite negative exponents as fractions and simplifies expressions. For example:

• \( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \)

Applying this to our exercise, we start with \( \frac{1}{t^5} \). To express it using negative exponents, we can rewrite t with a negative exponent, bringing the base to the numerator:

\[ \frac{1}{t^5} = t^{-5} \].

So, in summary, when you have a positive exponent in the denominator, you can move the base to the numerator by making the exponent negative, as shown: \( \frac{1}{t^5} \) becomes \( t^{-5} \).