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Chapter 12: Problem 86

Simplify the expression. $$\frac{8 x^{2}}{3} \cdot \frac{9}{16 x}$$

Short Answer

Expert verified
The simplification of the given expression results in \(\frac{3x}{2}\).

Step by step solution

01

Write Out the Expression

Rewrite the given expression. With it:\[\frac{8 x^{2}}{3} \cdot \frac{9}{16 x}\] the aim is to simplify it as much as possible using the standard rules of computation for rational expressions.
02

Simplify by Multiplication

Rewrite the expression by multiplying the numerators together and the denominators together: \[\frac{8 x^{2}\cdot 9}{3\cdot 16 x}\] simplifying further produces the expression \[\frac{72x^2}{48x}\].
03

Simplify by Cancelling

Next, simplify the expression by cancelling out common terms in the numerator and denominator. In this case, one 'x' in the numerator and the 'x' in the denominator can cancel out, and the coefficient can be simplified by dividing both the numerator and denominator by their greatest common factor (24 in this case) providing a simplified expression of \[\frac{3x}{2}\].

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Key Concepts

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

Multiplication of Fractions
When multiplying fractions, it's important to remember that the process involves both numerators and denominators. For a fraction
  • The numerator represents the top part of the fraction.
  • The denominator represents the bottom part.
To multiply fractions, follow these simple steps:
  • Multiply the numerators together.
  • Multiply the denominators together.
  • Place the resulting numerators over the resulting denominators to form a new fraction.
    • In our original exercise expression \[\frac{8 x^{2}}{3} \cdot \frac{9}{16 x}\],
    we multiply the numerators \(8x^2\) and \(9\) to get \(72x^2\),
    and the denominators \(3\) and \(16x\) to get \(48x\).
    This results in a new fraction \(\frac{72x^2}{48x}\). Keep practicing this technique, and soon you'll find it quite straightforward to multiply any fractions together.
    Cancelling Common Factors
    Cancelling common factors is akin to simplifying fractions. It involves identifying and removing common terms from the numerator and the denominator. This is crucial in rational expressions, as it simplifies the expression to its most basic form. The steps involve:
    • Examining both the numerator and denominator for similar elements (numbers or variables).
    • Dividing both by their common factors, simplifying the overall fraction.
    Considering the expression \(\frac{72x^2}{48x}\),
    we notice that one \(x\) in \(x^2\) (numerator) can be cancelled out by the \(x\) in the denominator.
    This gives us \(\frac{72x}{48}\).
    You now have a simpler fraction, but you still need to look for any numerical common factors to reduce the fraction further.
    Greatest Common Factor
    Finding the greatest common factor (GCF) is instrumental in reducing rational expressions to their simplest form. The GCF is the largest number that divides both the numerator and the denominator without leaving a remainder.
    To find the GCF, you can:
    • List the factors of each number.
    • Identify the largest factor that appears in both lists.
    In our expression \(\frac{72x}{48}\), we focus on the coefficients 72 and 48.
    Both 72 and 48 can be divided by 24, making it the GCF.
    Dividing both numerator and denominator by 24,
    we simplify \(\frac{72}{48}\) to \(\frac{3}{2}\).
    Don't forget to apply the same division rule to any variables that remain.
    This results in the fully simplified expression \(\frac{3x}{2}\).

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