Problem 86 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{-6 a^{8} y}{3 a^{4} y} $$

Step-by-Step Solution

Answer

The simplified expression is \(-2a^4\).

Step 1: Simplify the Coefficients

Start by simplifying the coefficients, which are the numeric parts of the fraction. Here, you have \(-6\) divided by \(3\). Divide \(-6\) by \(3\) to get \(-2\). Thus, the numeric part of the fraction becomes \(-2\).

Step 2: Simplify the Term with 'a'

Look at the terms containing \(a\). You have \(a^8\) in the numerator and \(a^4\) in the denominator. When you divide terms with the same base, you subtract the exponents: \(a^{8-4} = a^4\). This simplifies the \(a\) terms to \(a^4\).

Step 3: Simplify the Term with 'y'

Now, simplify the terms containing \(y\). You have \(y\) in the numerator and \(y\) in the denominator. Because \(y\div y = 1\), they cancel each other out, leaving no \(y\) terms in the simplified expression.

Step 4: Combine Simplified Parts

Combine all the simplified parts from the previous steps. The expression \(\frac{-6 a^{8} y}{3 a^{4} y}\) simplifies to \(-2a^4\) because both y terms cancelled out and you have simplified terms for the coefficients and the a's.

Key Concepts

Fraction Simplification

Simplifying fractions is all about reducing them to their simplest form. The goal is to make the fraction as simple as possible without changing its value.

• Start by looking at the numeric coefficients: the numbers in front of any variables. In our exercise, these are \(-6\) and \(3\).
• Divide the numerator by the denominator, just like with regular numbers. Here, \(-6/3\) simplifies to \(-2\).
• Now, the numeric part of the fraction becomes cleaner and easier to handle.

It is essential to simplify fractions whenever possible to make calculations more straightforward and to better understand the relationships between numbers.

Division of Exponents

Dividing exponents can seem tricky at first, but it's really a simple process once you get the hang of it. When you divide terms with the same base, you can subtract the exponents:

• For this exercise, focus on the terms \(a^8\) and \(a^4\). Both share the same base, \(a\).
• Take the exponent of the term in the numerator (\(8\) for \(a^8\))and subtract the exponent of the denominator (\(4\) for \(a^4\)).
• This subtraction gives you \(a^{8-4} = a^4\), resulting in the simplified form of \(a^4\).

Using this rule lets you quickly reduce the complexity of expressions involving powers. Always ensure the bases are the same before applying this rule.

Canceling Terms

Canceling terms is a crucial step in simplifying algebraic expressions. It involves removing equal terms that appear in both the numerator and the denominator:

• Look at the variable \(y\) in the given expression. Both the numerator and the denominator have one \(y\).
• When you have \(y/y\), it simplifies directly to \(1\), effectively removing \(y\) from the expression.

This step is often referred to as canceling terms. It's simple yet vital, as it dramatically simplifies the expression you're working with. Remember, cancel only identical terms; they disappear completely, leaving a cleaner, more manageable expression.