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Chapter 4: Problem 93

Reduce the given fraction to lowest terms. $$\frac{-8 x^{6} y^{3}}{54 x^{3} y^{5}}$$

Short Answer

Expert verified
The reduced fraction is \(\frac{-4x^3}{27y^2}\).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

To reduce the fraction \(\frac{-8x^6y^3}{54x^3y^5}\), we first need to find the GCF of the coefficients, which are -8 and 54. The GCF of 8 and 54 is 2.
02

Simplify Coefficients

Divide both the numerator and the denominator by the GCF. This gives us \(\frac{-4x^6y^3}{27x^3y^5}\).
03

Simplify the Variables

For the \(x\)-terms, subtract the exponent in the denominator from the exponent in the numerator: \(x^{6-3} = x^3\).For the \(y\)-terms, do the same: \(y^{3-5} = y^{-2}\), which can also be written as \(\frac{1}{y^2}\).
04

Combine Simplified Terms

After simplifying, the fraction becomes \(\frac{-4x^3}{27y^2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor (GCF)
When simplifying fractions, one of the key steps is finding the Greatest Common Factor, or GCF. The GCF is the largest number that evenly divides into each of the numbers you are working with. In this exercise, we looked at the coefficients -8 and 54.
To find the GCF, you need to break down each number into its prime factors:
  • -8 factors into 2, 2, 2 (-1 is not considered since only positive factors are needed for GCF).
  • 54 factors into 3, 3, 3, 2.
The common factor among these is 2. So, 2 is the GCF.
By dividing both the numerator and the denominator by the GCF, we effectively reduce the fraction, making it simpler without changing its value.
Exponents
Exponents are a way to express repeated multiplication of a number. They consist of a base number and an exponent. The exponent tells us how many times to multiply the base by itself. In the given exercise we have two variables with exponents: \(x^6\) and \(y^3\) in the numerator, and \(x^3\) and \(y^5\) in the denominator.
To simplify these, we apply the rule of subtracting exponents when dividing.
For \(x\):
  • Subtract the exponent of the denominator (3) from the numerator (6): \(x^{6-3} = x^3\).
For \(y\):
  • Subtract the exponent of the denominator (5) from the numerator (3): \(y^{3-5} = y^{-2}\). This can also be rewritten as \(\frac{1}{y^2}\) because a negative exponent indicates reciprocal or division.
Understanding and applying these exponent rules is crucial in simplifying expressions.
Rational Expressions
A rational expression is essentially a fraction where the numerator and the denominator are polynomials. Rational expressions can often be simplified by canceling common factors, similar to how we simplify numerical fractions.
In the exercise, we have \[\frac{-8x^6y^3}{54x^3y^5}\]which is a rational expression. The goal is to simplify the expression to its lowest terms.
We do this by:
  • Finding and dividing by the GCF of the coefficients: For -8 and 54, it was 2.
  • Cancelling the corresponding variables by applying exponent rules.
After simplification, we obtain a much simpler rational expression, \[\frac{-4x^3}{27y^2}\], that maintains the same value but is easier to work with and understand.

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