$\left(-\frac{5}{4}\right)$ $-\frac{9}{12}$ 1. Identify the base: $-\frac{3}{4}$ $-\frac{27}{64}$ 2. Determine the exponent: 3 $\frac{9}{12}$ 3. Write in expanded form: $\left(-\frac{3}{4}\right)\left(-\frac{3}{4}\right)\left(-\frac{3}{4}\right)$ $\frac{27}{64}$
Answer (1)
The base is − 4 3 and the exponent is 3.
The expanded form is ( − 4 3 ) × ( − 4 3 ) × ( − 4 3 ) .
Calculating the expression gives − 64 27 .
The final answer is − 64 27 .
Explanation
Understanding the Problem We are given an expression and asked to identify the base and exponent, write in expanded form, and simplify. The expression seems to be evaluating ( − 4 3 ) 3 . We need to verify the final result.
Identifying Base and Exponent The base is identified as − 4 3 and the exponent is 3. This means we need to multiply the base by itself three times: ( − 4 3 ) 3 = ( − 4 3 ) × ( − 4 3 ) × ( − 4 3 ) .
Calculating the Expanded Form Now, let's calculate the expanded form: ( − 4 3 ) × ( − 4 3 ) × ( − 4 3 ) = 4 × 4 × 4 ( − 3 ) × ( − 3 ) × ( − 3 ) = 64 − 27 = − 64 27 .
Determining the Final Value The final result should be − 64 27 . However, the problem states the final result as 64 27 , which is incorrect. The correct result, considering the negative sign, is − 64 27 .
Examples
Understanding exponents and negative signs is crucial in many real-world applications. For example, calculating compound interest involves exponents, and understanding how negative signs affect financial calculations is essential for managing debt or investments. Similarly, in physics, calculating the decay of radioactive materials uses exponential functions, where negative signs indicate a decrease in quantity over time. These concepts are also fundamental in computer science for representing and manipulating numerical data.
