Problem 8 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{-18 x^{3} y}{-36 x y^{3}}$$

Step-by-Step Solution

Answer

The simplified form is \(\frac{x^2}{2y^2}\).

Step 1: Factor the numbers

Look at the numerical coefficients \(-18\) and \(-36\). Notice that both are divisible by \(-18\). So, \[-18 = -18 \-36 = (-18) \times 2\]Divide both numbers by \(-18\): \[-18 / -18 = 1\-36 / -18 = 2\]Thus, \(-18x^3y / -36xy^3 = 1x^3y / 2xy^3\).

Step 2: Simplify the variables

We have \(x^3\) in the numerator and \(x\) in the denominator:\[\frac{x^3}{x} = x^{3-1} = x^2\]Now simplify the \(y\) terms. We have \(y\) in the numerator and \(y^3\) in the denominator:\[\frac{y}{y^3} = y^{1-3} = y^{-2}\]Therefore, combining these, we have:\[\frac{x^2}{2y^2}\].

Step 3: Combine simplified factors

Now put it all together from the simplification:We have \(\frac{x^3y}{2xy^3} = \frac{x^2}{2y^2}\).Thus, we can conclude the simplified form of the fraction is \[\frac{x^2}{2y^2}\].

Key Concepts

Simplifying Fractions

Simplifying algebraic fractions is a vital skill. It makes working with complex equations easier. When we simplify a fraction, we're trying to make it as simple as possible while keeping its value the same.
To begin simplifying, we look at both the numerator and the denominator of the fraction. We aim to reduce them to their most basic form by canceling out common factors.
Consider the coefficients first: if both the top and bottom numbers can be divided by the same number, do this division. For example, in the fraction \( \frac{-18x^3y}{-36xy^3} \), both \(-18\) and \(-36\) can be divided by \(-18\). By doing so, they become 1 and 2 respectively.
Next, we turn our attention to the variables, which involves using exponent rules to simplify further.

Factoring Coefficients

Factoring coefficients is the process of breaking down numbers into their basic constituents. This helps in simplifying fractions.
When both the numerator and the denominator have whole numbers, check for any common factors. For example, in \( -18 \) and \( -36 \), both numbers can be divided by \(-18\).
- Start by writing out the factors.
-18 can be factored as \(-18 \times 1\)
-36 as \(-18 \times 2\).
- Divide each by the greatest common factor, which is \(-18\) here.
So it becomes \( \frac{-18x^3y}{-36xy^3} = \frac{1x^3y}{2xy^3} \).
Factoring helps reveal the simpler components of each term, setting the stage for exponent simplification next.

Exponent Rules

Exponent rules simplify multiplication and division when variables have power indicators. They allow us to easily switch between different forms.
When dealing with exponents, similar bases can be combined. In the fraction \( \frac{1x^3y}{2xy^3} \), when simplifying, we apply the rules of exponents. Here’s how:
- For \( x^3 \) and \( x \), use the exponent rule \( a^m / a^n = a^{m-n} \).
This gives us \( x^{3-1} = x^2 \).
- Similarly, for \( y \) and \( y^3 \), apply the same rule: \( y^{1-3} = y^{-2} \).
So, the fraction ultimately simplifies to \( \frac{x^2}{2y^2}\).
Understanding and applying these exponent rules effectively reduces complexity in equations and makes solving easier.