- Simplify the numerical coefficient: \(\frac{28}{12} = \frac{7}{3}\).
- Apply the quotient rule of exponents to the variable \(p\): \(\frac{p^9}{p^{-6}} = p^{15}\).
- Apply the quotient rule of exponents to the variable \(q\): \(\frac{q^{-5}}{q^7} = q^{-12} = \frac{1}{q^{12}}\).
- Combine the results to get the final simplified expression: \(\boxed{\frac{7 p^{15}}{3 q^{12}}}\).
Explanation
- Understanding the Problem
We are tasked with simplifying the algebraic expression \(\frac{28 p^9 q^{-5}}{12 p^{-6} q^7}\), given that \(p
eq 0\) and \(q
eq 0\). To simplify this, we treat the numerical coefficients and the variables separately using the standard laws of exponents, specifically \(\frac{a^m}{a^n} = a^{m-n}\) and \(a^{-n} = \frac{1}{a^n}\).
- Simplifying the Coefficients
First, we focus on the numerical coefficients: \(\frac{28}{12}\). Both 28 and 12 are divisible by 4, so we can simplify the fraction:
\[\frac{28}{12} = \frac{28 \div 4}{12 \div 4} = \frac{7}{3}.\]
- Simplifying the Variables
Next, we simplify the powers of \(p\) and \(q\) using the quotient rule for exponents \(\frac{x^m}{x^n} = x^{m-n}\):
For \(p\):
\[\frac{p^9}{p^{-6}} = p^{9 - (-6)} = p^{9 + 6} = p^{15}.\]
For \(q\):
\[\frac{q^{-5}}{q^7} = q^{-5 - 7} = q^{-12}.\]
Using the definition of negative exponents, \(q^{-12} = \frac{1}{q^{12}}\).
- Conclusion
Finally, we combine all parts together to form the complete simplified expression:
\[\frac{28 p^9 q^{-5}}{12 p^{-6} q^7} = \left(\frac{7}{3}\right) \times p^{15} \times \frac{1}{q^{12}} = \frac{7 p^{15}}{3 q^{12}}.\]
Comparing this result to the given options, we identify that \(\frac{7 p^{15}}{3 q^{12}}\) is the correct expression.
Examples
Algebraic simplification is essential in fields like physics and engineering, where complex formulas often need to be reduced to their simplest form to be easier to work with or compute. For example, if you are calculating the force between two charged particles, you might end up with an expression involving various powers of distances and charges. Simplifying this allows you to quickly see how doubling a distance or tripling a charge will affect the total force. Masterfully handling these algebraic expressions allows you to communicate physical relationships clearly and accurately.