Divide (8x^8y^8)/(x^5y^5)

Solution:

Step 1

Eliminate the common $x$ terms from the numerator and denominator.

Step 1.1

Extract $x^5$ from $8x^8y^8$ to get $\frac{x^5(8x^3y^8)}{x^5y^5}$.

Step 1.2

Simplify by removing the common factors.

Step 1.2.1

Take $x^5$ out of $x^5y^5$ to form $\frac{x^5(8x^3y^8)}{x^5(y^5)}$.

Step 1.2.2

Eliminate the matching $x^5$ terms to obtain $\frac{\cancel{x^5}(8x^3y^8)}{\cancel{x^5}y^5}$.

Step 1.2.3

Reformulate the fraction as $\frac{8x^3y^8}{y^5}$.

Step 2

Remove the common $y$ terms from the numerator and denominator.

Step 2.1

Factor out $y^5$ from $8x^3y^8$ to get $\frac{y^5(8x^3y^3)}{y^5}$.

Step 2.2

Simplify by canceling out the common factors.

Step 2.2.1

Multiply by $1$ to maintain equality, resulting in $\frac{y^5(8x^3y^3)}{y^5 \cdot 1}$.

Step 2.2.2

Eliminate the matching $y^5$ terms to achieve $\frac{\cancel{y^5}(8x^3y^3)}{\cancel{y^5} \cdot 1}$.

Step 2.2.3

Reformulate the fraction as $\frac{8x^3y^3}{1}$.

Step 2.2.4

Divide $8x^3y^3$ by $1$ to finalize the result as $8x^3y^3$.

Knowledge Notes:

The problem involves simplifying a fraction where both the numerator and the denominator contain powers of variables. The process uses the following algebraic rules and concepts:

1. Exponent Laws: When dividing like bases with exponents, subtract the exponents: $x^a / x^b = x^{a-b}$.

2. Cancellation: When the same term appears in both the numerator and the denominator, it can be cancelled out.

3. Multiplying by 1: Multiplying a term by 1 does not change its value. This is often used to show that cancelling terms is valid, as it's equivalent to multiplying by a form of 1 (e.g., $x^5/x^5 = 1$).

4. Simplifying Fractions: A fraction is fully simplified when no further common factors exist between the numerator and the denominator.

5. Final Division by 1: Dividing any term by 1 leaves the term unchanged.

The solution process involves applying these rules step by step to simplify the given algebraic fraction to its simplest form.