Simplify $\sqrt{\frac{72 a^3 b^{-2}}{2 a^5 b^{-6}}}$
Answer (1)
Simplify the fraction inside the square root by dividing the coefficients and using exponent rules.
Apply the square root to each part of the simplified expression.
Simplify the square roots of the numerical coefficient and the variable terms.
The simplified expression is a 6 b 2 .
Explanation
Understanding the problem We are asked to simplify the expression 2 a 5 b − 6 72 a 3 b − 2 . This involves simplifying a fraction inside a square root using properties of exponents.
Simplifying the fraction First, simplify the fraction inside the square root: 2 a 5 b − 6 72 a 3 b − 2 = 2 72 ⋅ a 5 a 3 ⋅ b − 6 b − 2
Simplifying the numerical part Simplify the numerical part: 2 72 = 36
Simplifying the a part Simplify the a part: a 5 a 3 = a 3 − 5 = a − 2
Simplifying the b part Simplify the b part: b − 6 b − 2 = b − 2 − ( − 6 ) = b − 2 + 6 = b 4
Combining the parts Combine the simplified parts: 36 a − 2 b 4
Rewriting the expression Rewrite the expression inside the square root: 36 a − 2 b 4
Applying the square root Apply the square root: 36 a − 2 b 4 = 36 ⋅ a − 2 ⋅ b 4
Simplifying Simplify: 36 ⋅ a − 2 ⋅ b 4 = 6 ⋅ a − 1 ⋅ b 2 = a 6 b 2
Examples
In physics, when dealing with wave phenomena or quantum mechanics, you often encounter expressions involving square roots and exponents. Simplifying such expressions, like the one in this problem, allows for easier manipulation and interpretation of physical quantities. For instance, if 'a' represents a distance and 'b' represents a frequency, the simplified expression could relate to a characteristic property of a wave, making the calculation of its value more straightforward.
