Simplify the expression:
$\frac{2 p}{q^2} \times \frac{2^{-1} p^{-1}}{q^2}$
To simplify the expression \(\frac{2 p}{q^2} \times \frac{2^{-1} p^{-1}}{q^2}\), we first analyze the structure. We are multiplying two fractions, which can be done by multiplying the numerators together and the denominators together.
The expression is:
\[\frac{2 p}{q^2} \times \frac{2^{-1} p^{-1}}{q^2} = \frac{(2 \cdot p) \cdot (2^{-1} \cdot p^{-1})}{q^2 \cdot q^2}\]
We have variables \(p\) and \(q\), along with the constant \(2\). We will use exponent rules to simplify the terms.
Now we group the terms with the same base in the numerator and the denominator.
In the numerator, we have:
\[2 \cdot 2^{-1} \cdot p \cdot p^{-1}\]
Using the rule for exponents \(x^a \cdot x^b = x^{a+b}\), we can simplify these:
\[2^{1} \cdot 2^{-1} = 2^{1 + (-1)} = 2^0 = 1\]
\[p^{1} \cdot p^{-1} = p^{1 + (-1)} = p^0 = 1\]
So, the entire numerator simplifies to \(1 \cdot 1 = 1\).
Next, we simplify the denominator:
\[q^2 \cdot q^2\]
Again, using the exponent rule \(x^a \cdot x^b = x^{a+b}\):
\[q^2 \cdot q^2 = q^{2+2} = q^4\]
Combining the numerator and the denominator, the expression becomes:
\[\frac{1}{q^4}\]
This is the fully simplified form of the given expression.
In physics and engineering, simplified expressions are crucial for calculating physical quantities efficiently. For instance, if you are determining the force exerted by an object or the resistance in an electrical circuit, algebraic simplification allows you to reduce complex formulas to their simplest form, saving time and reducing potential errors during calculation. Mastering this skill is a fundamental step in solving more advanced problems in science and mathematics.