Simplify $\left(\frac{3}{8}\right)^{-2}\left(\frac{1}{3} \cdot \frac{3}{8}\right)^3\left(\frac{1}{3}\right)^4$.
Answer (1)
Simplify the expression inside the parenthesis: 3 1 ⋅ 8 3 = 8 1 .
Rewrite the negative exponent: ( 8 3 ) − 2 = ( 3 8 ) 2 = 9 64 .
Expand the remaining terms: ( 8 1 ) 3 = 512 1 and ( 3 1 ) 4 = 81 1 .
Multiply and simplify the fractions: 9 64 ⋅ 512 1 ⋅ 81 1 = 5832 1 .
The simplified expression is 5832 1 .
Explanation
Understanding the Problem We are asked to simplify the expression ( 8 3 ) − 2 ( 3 1 ⋅ 8 3 ) 3 ( 3 1 ) 4 using the properties of exponents. Let's break it down step by step!
Simplifying the Expression First, simplify the term inside the second parenthesis: 3 1 ⋅ 8 3 = 3 ⋅ 8 1 ⋅ 3 = 24 3 = 8 1 So the expression becomes: ( 8 3 ) − 2 ( 8 1 ) 3 ( 3 1 ) 4
Dealing with the Negative Exponent Next, we deal with the negative exponent. Remember that a − n = a n 1 . Therefore: ( 8 3 ) − 2 = ( 3 8 ) 2 = 3 2 8 2 = 9 64 Now the expression is: 9 64 ⋅ ( 8 1 ) 3 ⋅ ( 3 1 ) 4
Expanding the Terms Now, let's expand the remaining terms: ( 8 1 ) 3 = 8 3 1 3 = 512 1 ( 3 1 ) 4 = 3 4 1 4 = 81 1 So the expression becomes: 9 64 ⋅ 512 1 ⋅ 81 1
Multiplying the Fractions Now, multiply the fractions: 9 64 ⋅ 512 1 ⋅ 81 1 = 9 ⋅ 512 ⋅ 81 64 ⋅ 1 ⋅ 1 = 9 ⋅ 512 ⋅ 81 64 We can simplify this by noticing that 512 = 8 ⋅ 64 , so: 9 ⋅ ( 8 ⋅ 64 ) ⋅ 81 64 = 9 ⋅ 8 ⋅ 81 1 Now, multiply the numbers in the denominator: 9 ⋅ 8 ⋅ 81 = 72 ⋅ 81 = 5832 So the simplified expression is: 5832 1
Final Answer Therefore, the simplified form of the given expression is 5832 1 .
Examples
Understanding and simplifying expressions with exponents is crucial in various fields, such as physics and engineering, where exponential growth and decay are common. For example, calculating the decay rate of a radioactive substance or determining the growth of a bacterial colony involves simplifying exponential expressions. This skill also helps in financial mathematics when dealing with compound interest calculations, where the principal amount grows exponentially over time.
