Simplify $\left(1.9^2 \cdot 2.4^{-3}\right)^2\left(1.9^3 \cdot 2.4^{-2}\right)^{-3}$.
Answer (2)
Apply the power of a product and power of a power rules to get ( 1. 9 2 ⋅ 2. 4 − 3 ) 2 ( 1. 9 3 ⋅ 2. 4 − 2 ) − 3 = ( 1. 9 4 ⋅ 2. 4 − 6 ) ( 1. 9 − 9 ⋅ 2. 4 6 ) .
Combine like terms by adding exponents: 1. 9 4 − 9 ⋅ 2. 4 − 6 + 6 = 1. 9 − 5 ⋅ 2. 4 0 .
Simplify 2. 4 0 to 1, resulting in 1. 9 − 5 ⋅ 1 = 1. 9 − 5 .
The simplified expression is 1. 9 − 5 .
Explanation
Understanding the Problem We are asked to simplify the expression ( 1. 9 2 ⋅ 2. 4 − 3 ) 2 ( 1. 9 3 ⋅ 2. 4 − 2 ) − 3 . This involves using the rules of exponents to combine terms with the same base.
Applying Exponent Rules First, we apply the power of a product rule, ( ab ) n = a n b n , and the power of a power rule, ( a m ) n = a mn , to each term in the expression: \begin{align*} \left(1.9^2 \cdot 2.4^{-3}\right)^2 &= (1.9^2)^2 \cdot (2.4^{-3})^2 = 1.9^{2\cdot 2} \cdot 2.4^{-3\cdot 2} = 1.9^4 \cdot 2.4^{-6} \ \left(1.9^3 \cdot 2.4^{-2}\right)^{-3} &= (1.9^3)^{-3} \cdot (2.4^{-2})^{-3} = 1.9^{3\cdot (-3)} \cdot 2.4^{-2\cdot (-3)} = 1.9^{-9} \cdot 2.4^{6} \end{align*}
Substituting Back Now, we substitute these simplified terms back into the original expression:
( 1. 9 4 ⋅ 2. 4 − 6 ) ⋅ ( 1. 9 − 9 ⋅ 2. 4 6 )
Combining Like Terms Next, we combine the terms with the same base by adding their exponents, using the rule a m ⋅ a n = a m + n :
1. 9 4 ⋅ 1. 9 − 9 ⋅ 2. 4 − 6 ⋅ 2. 4 6 = 1. 9 4 + ( − 9 ) ⋅ 2. 4 − 6 + 6 = 1. 9 − 5 ⋅ 2. 4 0
Simplifying Further Since any number raised to the power of 0 is 1 (i.e., a 0 = 1 ), we have 2. 4 0 = 1 . Therefore, the expression simplifies to:
1. 9 − 5 ⋅ 1 = 1. 9 − 5
Final Simplification Finally, we can rewrite 1. 9 − 5 as 1. 9 5 1 . Calculating 1. 9 5 gives approximately 24.76099. Therefore,
1. 9 − 5 = 1. 9 5 1 = 24.76099 1 ≈ 0.040386
However, the question asks for the simplified expression, so we leave it as 1. 9 − 5 .
Final Answer Thus, the simplified expression is 1. 9 − 5 .
Examples
Understanding how to simplify expressions with exponents is useful in many areas, such as calculating compound interest, where the initial investment grows exponentially over time. For example, if you invest P dollars at an annual interest rate r compounded n times per year for t years, the final amount A is given by A = P ( 1 + n r ) n t . Simplifying such expressions helps in financial planning and understanding growth rates.
The expression simplifies to 1. 9 − 5 using the power of products and powers of powers rules in exponents. Combining like terms leads to this final form, which captures the essence of the original expression. For clarity, it can also be expressed as 1. 9 5 1 .
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