Use laws of exponents to simplify inside the parentheses of the given expression.
$\left(\frac{a^{-2} b^2}{a^2 b^{-1}}\right)^{-3}$
Answer (1)
Simplify ( a 2 b − 1 a − 2 b 2 ) − 3 to b 9 a 12 .
Simplify b − 3 to b 3 1 .
Simplify ( a 4 b ) − 3 to b 3 a 12 .
Simplify ( b 3 ) − 3 to b 9 1 .
Simplify ( a 4 b 3 ) − 3 to b 9 a 12 .
Explanation
Review of Exponent Rules We will simplify each expression using the laws of exponents. The key rules we'll use are:
Quotient Rule: a n a m = a m − n
Power of a Quotient Rule: ( b a ) n = b n a n
Power of a Power Rule: ( a m ) n = a mn
Negative Exponent Rule: a − n = a n 1
Simplifying the First Expression (Inside Parentheses) First expression: ( a 2 b − 1 a − 2 b 2 ) − 3 .
Simplify inside the parentheses:
a 2 b − 1 a − 2 b 2 = a − 2 − 2 b 2 − ( − 1 ) = a − 4 b 3
Simplifying the First Expression (Outer Exponent) Now apply the outer exponent:
( a − 4 b 3 ) − 3 = a − 4 ×− 3 b 3 ×− 3 = a 12 b − 9 = b 9 a 12
Simplifying the Second Expression Second expression: b − 3 .
Using the negative exponent rule:
b − 3 = b 3 1
Simplifying the Third Expression Third expression: ( a 4 b ) − 3 .
Using the power of a quotient rule and the negative exponent rule:
( a 4 b ) − 3 = a − 4 × 3 b − 3 = a − 12 b − 3 = b 3 a 12
Simplifying the Fourth Expression Fourth expression: ( b 3 ) − 3 .
Using the power of a power rule:
( b 3 ) − 3 = b 3 ×− 3 = b − 9 = b 9 1
Simplifying the Fifth Expression Fifth expression: ( a 4 b 3 ) − 3 .
Using the power of a quotient rule and the negative exponent rule:
( a 4 b 3 ) − 3 = a 4 ×− 3 b 3 ×− 3 = a − 12 b − 9 = b 9 a 12
Final Simplified Expressions In summary:
( a 2 b − 1 a − 2 b 2 ) − 3 = b 9 a 12
b − 3 = b 3 1
( a 4 b ) − 3 = b 3 a 12
( b 3 ) − 3 = b 9 1
( a 4 b 3 ) − 3 = b 9 a 12
Examples
Understanding and simplifying expressions with exponents is crucial in many fields, including physics and engineering. For example, in electrical engineering, impedance calculations often involve complex numbers raised to various powers. Simplifying these expressions allows engineers to analyze circuits more efficiently and design systems that meet specific performance criteria. Similarly, in physics, understanding exponential relationships is essential for modeling phenomena such as radioactive decay or the behavior of gases under pressure.
