Verified Answer :

  • Rewrite the expression \(8p\) as \(\frac{8p(p+9)}{p+9}\).
  • Subtract the rational term \(\frac{8p}{p+9}\) from the rewritten first term.
  • Combine the numerators to obtain \(\frac{8p^2 + 72p - 8p}{p+9}\).
  • Simplify the final fraction to \(\boxed{\frac{8p^2+64p}{p+9}}\) or \(\boxed{\frac{8p(p+8)}{p+9}}\).

Explanation

  1. Analysis and Common Denominator

To simplify the expression \(8p - \frac{8p}{p+9}\), we first note that the expression is defined for all values of \(p\) such that \(p eq -9\). We aim to represent this expression as a single rational fraction by finding a common denominator for the two terms. The first term, \(8p\), can be written as \(\frac{8p}{1}\). The second term has a denominator of \(p+9\). Therefore, the common denominator for the entire expression is \(p+9\).

  1. Rewriting with a Common Denominator

We now express \(8p\) with the common denominator \(p+9\):
\[8p = \frac{8p \times (p+9)}{p+9} = \frac{8p^2 + 72p}{p+9}\]
Now we subtract the second term from this:
\[8p - \frac{8p}{p+9} = \frac{8p^2 + 72p}{p+9} - \frac{8p}{p+9}\]

  1. Final Simplification

Subtracting the two fractions gives us:
\[\frac{8p^2 + 72p - 8p}{p+9}\]
Combining the like terms in the numerator (specifically \(72p\) and \(-8p\)):
\[\frac{8p^2 + 64p}{p+9}\]
We can factor out \(8p\) from the numerator to express the result in a potentially simpler or more compact form:
\[\frac{8p(p + 8)}{p+9}\]
This is the single, simplest form of the rational expression.

Examples

In engineering or physics, simplified algebraic expressions often represent the total resistance of a circuit or the net force acting on an object. For instance, if you have a base load \(8p\) and a portion of it that is reduced by a ratio involving the variable \(p+9\), simplifying the expression helps you quickly calculate the effective value for various parameters of \(p\). This reduction of complex fractions into a single term streamlines calculations in design and analysis.

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