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Reducing fractions means simplifying them to their smallest form. This involves two main steps:

• Simplifying the numerical (coefficients) part of the fraction.
• Simplifying the variable part of the fraction.

Let's take the fraction from the exercise: \(\frac{6 a^{6}}{12 a^{12}}\).
First, focus on the coefficients. Divide the numerator by the denominator: \(\frac{6}{12}\), which simplifies to \(\frac{1}{2}\). Now the fraction looks like \(\frac{1 a^{6}}{2 a^{12}}\)
Next, simplify the variables with exponents. Use the rule that when dividing with the same base, subtract the exponents: \( \frac{a^{6}}{a^{12}} = a^{6-12} = a^{-6}\).
Now, putting everything together, you get the reduced fraction: \( \frac{1}{2 a^{6}} \).
Remember, a fraction is fully simplified when both its numerator and denominator are reduced to their smallest possible form.

Algebraic fractions have variables in the numerator, the denominator, or both. For example, in the fraction \(\frac{6 a^{6}}{12 a^{12}}\), both parts have the variable \(a\).
Working with algebraic fractions involves:

• Factorizing the expressions if necessary.
• Simplifying variables by using exponent rules.

Let's break it down with our example:
Start with the fraction \(\frac{6 a^{6}}{12 a^{12}}\). Simplify the coefficients first: \(\frac{6}{12} = \frac{1}{2}\).
Now simplify the algebraic part: subtract the exponents: \( \frac{a^{6}}{a^{12}} = a^{6-12} = a^{-6} \).
Combining these, you get: \( \frac{1}{2 a^{6}} \).
You simplified an algebraic fraction! Make sure to apply exponent rules and always check the coefficients and variables separately.

Exponent rules are key when simplifying algebraic fractions.
Here are a few important ones:

• \(a^m \times a^n = a^{m+n}\)
• \(\frac{a^m}{a^n}= a^{m-n}\) (used in the exercise)
• \((a^m)^n = a^{m\times n}\)

When you see variables with exponents, apply these rules to simplify. In the exercise \(\frac{6 a^{6}}{12 a^{12}}\):
First, simplify the coefficients: \(\frac{6}{12} = \frac{1}{2}\).
Then handle the exponents: \( \frac{a^{6}}{a^{12}} = a^{6-12} = a^{-6} \).
This makes the fraction \( \frac{1}{2 a^{6}} \).
Always remember to subtract the exponents when dividing like bases. This helps in reducing fractions with variables effectively.