Simplify $\left(\frac{2}{3}\right)^3\left(\frac{1}{5} \cdot \frac{2}{3}\right)^{-3}\left(\frac{1}{5}\right)^4$.
Answer (1)
Rewrite the expression using properties of exponents.
Simplify the expression by canceling out common factors.
Rewrite 1 5 3 as ( 3 ⋅ 5 ) 3 = 3 3 ⋅ 5 3 .
The simplified expression is 5 1 .
Explanation
Understanding the Problem We are asked to simplify the expression ( 3 2 ) 3 ( 5 1 ⋅ 3 2 ) − 3 ( 5 1 ) 4 using the properties of exponents.
Rewriting the Expression First, let's rewrite the expression as ( 3 2 ) 3 ( 15 2 ) − 3 ( 5 1 ) 4 .
Using Negative Exponents Next, we can rewrite ( 15 2 ) − 3 as ( 15 2 ) 3 1 = 1 5 3 2 3 1 = 2 3 1 5 3 .
Substituting Back Now, the expression becomes ( 3 2 ) 3 ⋅ 2 3 1 5 3 ⋅ ( 5 1 ) 4 .
Expanding the Terms We can rewrite ( 3 2 ) 3 as 3 3 2 3 and ( 5 1 ) 4 as 5 4 1 . So the expression is 3 3 2 3 ⋅ 2 3 1 5 3 ⋅ 5 4 1 .
Canceling Common Factors We can cancel out 2 3 from the numerator and the denominator, which gives us 3 3 ⋅ 5 4 1 5 3 .
Rewriting 15 Since 15 = 3 ⋅ 5 , we can rewrite 1 5 3 as ( 3 ⋅ 5 ) 3 = 3 3 ⋅ 5 3 . So the expression becomes 3 3 ⋅ 5 4 3 3 ⋅ 5 3 .
Canceling Common Factors Again Now, we can cancel out 3 3 from the numerator and the denominator, which gives us 5 4 5 3 .
Simplifying the Expression Finally, we simplify 5 4 5 3 to 5 1 .
Final Answer Therefore, the simplified expression is 5 1 .
Examples
Understanding exponents is crucial in many real-world scenarios, such as calculating compound interest. For example, if you invest P dollars at an annual interest rate r compounded n times per year, the amount A you'll have after t years is given by A = P ( 1 + n r ) n t . Simplifying expressions with exponents helps in determining the final amount more efficiently. Another example is in computer science, where memory sizes and processing speeds are often expressed in powers of 2, requiring a solid understanding of exponential properties.
