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Negative exponents can be a bit confusing at first, but they follow a simple rule. When you see a negative exponent, you can convert it into a positive exponent by moving the base to the opposite side of the fraction.
An exponent like \( c^{-5} \) in the denominator can be converted by moving it to the numerator and changing the sign of the exponent to positive. Thus, \( \frac{b^{2}}{c^{-5}} \) becomes \( b^{2} \times c^{5} \).
In general, remember the rule: For any non-zero number 'a' and positive integer 'n', \[ \frac{1}{a^{-n}} = a^{n} \]
Let's look at another example to reinforce this idea: \[ \frac{3^{-2}}{5^{3}} = \frac{1}{3^{2} \times 5^{3}} \]
This makes simplifying expressions much easier!

Positive exponents show us how many times a number (base) is multiplied by itself. For example, \[ b^{2} = b \times b \]
Combining positive exponents follows basic exponent rules. When multiplying two terms with the same base, you add the exponents together. For example: \[ x^{2} \times x^{3} = x^{2+3} = x^{5} \]
However, in our original exercise, the bases are different (b and c), so they do not combine. The final expression \( b^{2}c^{5} \) features positive exponents, making the expression as simplified as possible.
Keep in mind:

• Add exponents when multiplying with the same base: \[ a^m \times a^n = a^{m+n} \]
• Subtract exponents when dividing with the same base: \[ a^m / a^n = a^{m-n} \]

These rules apply for positive exponents and help simplify complex expressions.

Simplifying fractions is important in making equations easier to work with. When simplifying, follow these steps:

• **Identify** common factors in the numerator and denominator.
• **Cancel out** common factors.
• **Rewrite** the expression in its simplest form.

In our exercise, we converted the term with the negative exponent to the numerator, which resulted in \( b^{2} \times c^{5} \). Note that after this step, there are no common factors between the terms.
Keep practicing with different problems to become comfortable with simplifying fractions. Aim for expressions where each exponential term is straightforward and positive. Here's another fraction simplification example to illustrate further:
Let's simplify \[ \frac{6x^{3}}{2x} \]
First, divide numbers: \[ \frac{6}{2} = 3 \]
Second, subtract exponents: \[ x^{3-1}= x^{2} \]
So, \[ \frac{6x^{3}}{2x} = 3x^{2} \]
This method allows you to handle more complex expressions effectively!