Simplify: [tex]$\sqrt{\frac{245 p^2 q^3 r^5}{7 p^4 q^5 r^7}}$[/tex]
Answer (2)
Simplify the fraction inside the square root: 7 245 = 35 .
Simplify the variable exponents: p 4 p 2 = p − 2 , q 5 q 3 = q − 2 , r 7 r 5 = r − 2 .
Rewrite the expression inside the square root: p 2 q 2 r 2 35 .
Take the square root: p 2 q 2 r 2 35 = pq r 35 .
The final simplified expression is pq r 35 .
Explanation
Understanding the Problem We are given the expression 7 p 4 q 5 r 7 245 p 2 q 3 r 5 and we want to simplify it. We will simplify the fraction inside the square root first, and then take the square root. We assume that p , q , r = 0 to avoid division by zero. We also assume that 0"> p , q , r > 0 to ensure that the expression is real and well-defined.
Simplifying the Fraction First, we simplify the fraction 7 245 , which equals 35. So we have p 4 q 5 r 7 35 p 2 q 3 r 5 .
Simplifying the Variables Next, we simplify the powers of p , q , and r . We have:
p 4 p 2 = p 2 − 4 = p − 2
q 5 q 3 = q 3 − 5 = q − 2
r 7 r 5 = r 5 − 7 = r − 2
Taking the Square Root So the expression inside the square root becomes p 2 q 2 r 2 35 . Now we take the square root of this expression:
p 2 q 2 r 2 35 = p 2 q 2 r 2 35 = pq r 35
Final Answer Since 35 = 5 * 7, and neither 5 nor 7 are perfect squares, 35 cannot be simplified further. Therefore, the final simplified expression is pq r 35 .
Examples
Imagine you are calculating the impedance of an electrical circuit, and you end up with a complex expression involving square roots and variables. Simplifying such expressions, as we did here, allows you to more easily understand and work with the circuit parameters. This type of simplification is also useful in physics when dealing with quantities like kinetic energy or gravitational potential energy, where you often need to simplify expressions to make calculations easier and more insightful.
The expression 7 p 4 q 5 r 7 245 p 2 q 3 r 5 simplifies to pq r 35 . The simplification was achieved by simplifying the fraction and the variable exponents before taking the square root. The final result is pq r 35 .
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