91Ó°ÊÓ

When we talk about simplifying fractions, we mean reducing them to their simplest form. This involves finding a common factor, which is a number that divides both the numerator (top number) and the denominator (bottom number) without leaving a remainder.

In our exercise, the coefficients numeric part of the fraction is \frac{45}{-60}. We need to find the Greatest Common Divisor (GCD) of 45 and -60 to simplify this fraction. The GCD of 45 and -60 is 15.

This means both 45 and -60 can be perfectly divided by 15. Simplifying the fraction by dividing both the numerator and the denominator by their GCD, we have:

\[ \frac{45}{-60}= \frac{45\div15}{-60\div15}= \frac{3}{-4} \]. This reduces the fraction substantially.

Exponents are a shorthand notation to show how many times a number, known as the base, is multiplied by itself.
For example, \(x^5\) means that x is multiplied by itself 5 times.

In our exercise, we need to divide variables with exponents. We have \(x^5\) divided by \(x^8\) and \(y^9\) divided by \(y^6\). To simplify these, we use the rule of dividing exponents:\( \frac{x^a}{x^b} = x^{a-b} \).

For the given problem:
\[ \frac{x^5}{x^8}= x^{5-8}=x^{-3} \]

Similarly for y:
\[ \frac{y^9}{y^6}= y^{9-6}= y^{3} \].

Combining these simplified outcomes with the numeric fraction simplified earlier, we form a part of our final answer.

The Greatest Common Divisor (GCD), also known as the greatest common factor, is the largest positive integer that divides each of the given integers without leaving a remainder. Finding the GCD is crucial when simplifying fractions.

For our given problem, the coefficients are 45 and -60. Using the Euclidean algorithm or prime factorization, we find that 15 is the GCD of 45 and -60. This allows us to simplify the fraction by dividing both the numerator and the denominator by this number.

Applying the GCD:
\[ \frac{45}{-60} = \frac{3}{-4} \].

Understanding and applying the concept of GCD helps in breaking down fractions to their simplest form, making the calculations more manageable and the results clearer.