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When we talk about simplifying fractions, we refer to reducing the fraction to its lowest terms. This means that the numerator (the top number) and the denominator (the bottom number) have no common factors other than 1. To simplify a fraction:

• Identify the greatest common divisor (GCD) of both the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

In our exercise, we simplify \( \frac{30}{25t^{10}} \) by first finding the GCD of 30 and 25, which is 5. Dividing both by 5 gives us \( \frac{6}{5t^{10}} \), which is the fraction in its simplest form. Remember, the goal is always to make the fraction as straightforward as possible, enabling easier subsequent mathematical operations.

Multiplication involving negative numbers can sometimes confuse students, but the rule is straightforward. When you multiply two negative numbers, the result is always positive. In contrast, if you multiply a positive number with a negative one, the result is negative.

In the given problem, we multiply \( -\frac{10}{t^{7}} \) with \( -\frac{3 t^{2}}{25 t} \). Here, the numerators -10 and -3 are both negative. According to the rule of negative number multiplication, their product will be positive, resulting in a positive 30. This rule is crucial as it affects the sign of your answer and can change the direction of inequalities in an equation. Always keep this rule in mind when dealing with negative values.

Understanding exponent rules is essential when manipulating expressions involving powers. The rule of multiplying exponents states that when you multiply two powers with the same base, you add the exponents. However, if the base is not the same, you cannot combine the exponents in this way.

For example, \(t^{7} * t^{2} \) becomes \(t^{7+2} \) or \(t^{9} \). In our exercise, the denominators were \(t^{7} \) and \(t^{2} \) that combined to \(t^{9} \). However, there's also \(25t \) in the denominator, which means we actually have \(t^{9} * t^{1} \) or \(t^{10} \). Understanding the exponent rules can simplify many algebraic expressions and is fundamental to algebra.