91Ó°ÊÓ

Simplifying algebraic fractions is a crucial skill in algebra. It involves reducing a fraction to its simplest form.
To simplify a fraction, follow these steps:

• Divide the numerator and the denominator by their greatest common divisor.
• Cancel out any common factors that appear in both the numerator and the denominator.

Consider the original problem:
ewline \(\frac{18 x^{3} y^{2}}{7 z^{4}} \div(14 x^{2} y^{2} z^{2})\).
First, we rewrote the division as multiplication by the reciprocal:
\(\frac{18 x^{3} y^{2}}{7 z^{4}} \times \frac{1}{14 x^{2} y^{2} z^{2}}\).
We then combined the numerators and denominators:
\(\frac{18 x^{3} y^{2}}{98 x^{2} y^{2} z^{6}}\).
Next, we simplified the coefficients by dividing them:
\(\frac{9 x^{3} y^{2}}{49 x^{2} y^{2} z^{6}}\).
Finally, we simplified the variables by canceling out common terms:
\(\frac{9 x^{1}}{49 z^{6}}\).
This leads us to the final simplified form:
\(\frac{9 x}{49 z^{6}}\).

Handling variable exponents in algebraic fractions requires understanding the rules of exponents.
Here are essential rules:

• When multiplying like bases, add the exponents: \(a^m \times a^n = a^{m+n}\).
• When dividing like bases, subtract the exponents: \(a^m \div a^n = a^{m-n}\).

In our example, after the fractions were combined, we had:
\(\frac{18 x^{3} y^{2} \times 1}{98 x^{2} y^{2} z^{6}}\).
We noticed common variables with exponents in the numerator and the denominator.
To simplify, we used the rule for division of like bases:
Subtract the exponents of \(x\) and \(y\):
{\(x^{3-2}\)} and {\(y^{2-2}\)}, leading to:
\(\frac{9 x^{1}}{49 z^{6}}\).
This creates a simpler fraction with reduced exponents.

Multiplying algebraic fractions is straightforward if you follow the steps.
Here is a quick process:

• Multiply the numerators together.
• Multiply the denominators together.
• Simplify the resulting fraction.

In our exercise, we started with:
\(\frac{18 x^{3} y^{2}}{7 z^{4}} \times \frac{1}{14 x^{2} y^{2} z^{2}}\).
We multiplied the numerators:
\(18 x^{3} y^{2} \times 1 = 18 x^{3} y^{2}\).
Then, we multiplied the denominators:
\(7 z^{4} \times 14 x^{2} y^{2} z^{2} = 98 x^{2} y^{2} z^{6}\).
Our fraction then looked like:
\(\frac{18 x^{3} y^{2}}{98 x^{2} y^{2} z^{6}}\).
With everything multiplied, we moved on to simplifying.