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Chapter 6: Problem 29

Reduce each fraction to simplest form. $$\frac{18 x^{2} y}{24 x y}$$

Short Answer

Expert verified
The simplest form of the fraction is \(\frac{3x}{4}\).

Step by step solution

01

Find the Greatest Common Factor (GCF)

Identify the greatest common factor of the numbers 18 and 24 and the lowest power of each variable in the expression. The number's GCF is 6. For the variables, the lowest powers are \(x^1\) and \(y^1\).
02

Divide by the GCF

Divide both the numerator and the denominator by their greatest common factor, including variables. Divide 18 and 24 by their GCF, which is 6. Also, divide the variable parts by \(x^1y^1\): \[\frac{18x^2y}{24xy} = \frac{18\div6 \cdot x^{2-1} \cdot y^{1-1}}{24\div6}\]
03

Simplify the Expression

Perform the division for each part of the fraction to get the simplified form:\[\frac{3x^{1}y^{0}}{4}\]Note that \(y^0 = 1\), simplifying further to \(\frac{3x}{4}\).

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Key Concepts

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

Understanding Greatest Common Factor
The Greatest Common Factor (GCF) is an essential concept when simplifying fractions. It is the largest number that divides two or more numbers without leaving a remainder. To determine the GCF of numbers like 18 and 24, you can list their factors and find the highest factor they have in common. For example:
  • Factors of 18: 1, 2, 3, 6, 9, 18
  • Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
The largest common factor is 6, the GCF.
When simplifying algebraic fractions, we also consider variable factors. If given \(x^2y\) and \(xy\), identify the lowest power of each variable to include in the GCF, which here is \(x^1y^1\). Finding the GCF helps in reducing fractions efficiently without missing any shared factors.
Simplifying Rational Expressions
Rational expressions are fractions where the numerator and the denominator are polynomials. Simplifying these expressions means reducing them to their simplest form, just like ordinary fractions. Consider rational expressions with both numeric and variable components:
  • The numeric part is reduced by dividing both by their GCF.
  • The variable parts are simplified by subtracting the powers of common variables, similar to exponent rules.
Take \(\frac{18x^2y}{24xy}\). First, reduce the numeric components by dividing 18 and 24 by their GCF, which is 6. Then apply the rules of exponents for the variable parts to process \(x^2/x\) and \(y/y\). This method effectively transforms the complex fraction into a simpler one.
Algebraic Simplification Techniques
Algebraic simplification involves reducing expressions to their simplest forms. This process makes them easier to work with and understand.
Simplifying expressions like \(\frac{18x^2y}{24xy}\) requires two main techniques:
  • Cancel out matching factors in the numerator and denominator.
  • Use exponent rules: Subtract exponents with the same base as in \(x^2/x\) simplifying to \(x\), and \(y^1/y^1\) simplifying to 1.
After applying these techniques, simplify further if possible. For example, when \(y^0\) equals 1, the \(y\) term disappears, leaving you with \(\frac{3x}{4}\). This final step indicates you have fully simplified the expression.

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Most popular questions from this chapter

Perform the indicated operations. Each expression occurs in the indicated area of application. $$\frac{\frac{L}{C}+\frac{R}{s C}}{s L+R+\frac{1}{s C}} \text { (electronics) }$$

Solve the given equations and check the results. $$\frac{x}{2}+6=2 x$$

Perform the indicated operations. Each expression occurs in the indicated area of application. $$\frac{b}{x^{2}+y^{2}}-\frac{2 b x^{2}}{x^{4}+2 x^{2} y^{2}+y^{4}} \text { (magnetic field) }$$

Solve the given equations and check the results. $$\frac{5}{2 n+4}=\frac{3}{4 n}$$

Solve the given equations and check the results. $$\frac{1}{4}-\frac{t-3}{8}=\frac{1}{6}$$
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