Which of the following is equal to $\left[\frac{\left(x^2 y^3\right)^{-2}}{\left(x^6 y^3 z\right)^2}\right]^3$?
A. $\frac{\left(x^2 y^3\right)^{-6}}{\left(x^6 y^3 z\right)^6}$
B. $\frac{1}{x^{48} y^{36} z^6}$
C. $\frac{\left(x^2 y^3\right)}{\left(x^6 y^3 z\right)^5}$
D. $\frac{x^{-12} y^{-18}}{x^{36} y^{18} z^8}$
Answer (1)
Simplify the inner expression using power of a power and power of a product rules.
Combine like terms by subtracting exponents.
Apply the outer exponent using the power of a product rule.
Rewrite the expression with positive exponents: x 48 y 36 z 6 1
Explanation
Understanding the Problem We are given the expression [ ( x 6 y 3 z ) 2 ( x 2 y 3 ) − 2 ] 3 . Our goal is to simplify this expression and find an equivalent expression from the given options.
Simplifying the Inner Expression First, let's simplify the inner expression by applying the power of a power rule, ( a m ) n = a mn , and the power of a product rule, ( ab ) n = a n b n :
[ ( x 6 y 3 z ) 2 ( x 2 y 3 ) − 2 ] 3 = [ x 12 y 6 z 2 x − 4 y − 6 ] 3
Combining Like Terms Next, we simplify the fraction inside the brackets by dividing terms with the same base. Recall that a n a m = a m − n .
[ x 12 y 6 z 2 x − 4 y − 6 ] 3 = [ x − 4 − 12 y − 6 − 6 z − 2 ] 3 = [ x − 16 y − 12 z − 2 ] 3
Applying the Outer Exponent Now, we apply the power of a product rule again and the power of a power rule to the entire expression: [ x − 16 y − 12 z − 2 ] 3 = x − 16 × 3 y − 12 × 3 z − 2 × 3 = x − 48 y − 36 z − 6
Rewriting with Positive Exponents Finally, we can rewrite the expression with positive exponents by using the rule a − n = a n 1 :
x − 48 y − 36 z − 6 = x 48 y 36 z 6 1
Finding the Equivalent Expression Comparing our simplified expression with the given options, we find that it matches the second option: x 48 y 36 z 6 1
Final Answer Therefore, the expression [ ( x 6 y 3 z ) 2 ( x 2 y 3 ) − 2 ] 3 is equal to x 48 y 36 z 6 1 .
Examples
Understanding how to simplify expressions with exponents is crucial in many fields, including physics and computer science. For example, in physics, when dealing with gravitational forces or electromagnetic fields, you often encounter expressions with exponents. Simplifying these expressions allows for easier calculations and a better understanding of the relationships between different variables. In computer science, exponents are used in algorithms for data compression and encryption, where efficient simplification can lead to faster and more secure processes. This skill is also essential in engineering for designing structures and systems that can withstand various forces and stresses.
