Simplify (76x^8)/(79x^4)

Solution:

Step 1:

Extract $x^{4}$ from the numerator $76x^{8}$ to get $\frac{x^{4}(76x^{4})}{79x^{4}}$.

Step 2:

Eliminate the common factors.

Step 2.1:

Extract $x^{4}$ from the denominator $79x^{4}$ to get $\frac{x^{4}(76x^{4})}{x^{4}\cdot 79}$.

Step 2.2:

Remove the common $x^{4}$ factor to simplify $\frac{\cancel{x^{4}}(76x^{4})}{\cancel{x^{4}}\cdot 79}$.

Step 2.3:

Present the simplified expression as $\frac{76x^{4}}{79}$.

Knowledge Notes:

When simplifying algebraic fractions, the goal is to reduce the expression to its simplest form by eliminating common factors from the numerator and the denominator.

1. Factorization: This involves breaking down a number or expression into its constituent factors. In this case, we factor $x^{4}$ out of both the numerator and the denominator.

2. Cancellation: If the same factor appears in both the numerator and the denominator, it can be cancelled out. This is based on the property that a fraction with the same non-zero number or expression in both the numerator and the denominator is equivalent to 1.

3. Exponent Rules: When dividing powers with the same base, you subtract the exponents. For example, $x^{m} / x^{n} = x^{m-n}$, provided $x$ is not zero.

4. Simplification: The process of simplification involves rewriting an expression in the most concise and straightforward manner without changing its value. After cancelling common factors, the remaining expression is the simplified form.

In the given problem, we use these principles to simplify the algebraic fraction $(76x^{8})/(79x^{4})$ by factoring out $x^{4}$ and cancelling it, leaving the simplified result of $(76x^{4})/79$.