Which expression has the greatest value?
$(1+2)^3$
$(2 \times 4)^1$
$(5-2)^2$
$(15-1)^0$
Answer (1)
Evaluate each expression: ( 1 + 2 ) 3 = 27 , ( 2 × 4 ) 1 = 8 , ( 5 − 2 ) 2 = 9 , ( 15 − 1 ) 0 = 1 .
Compare the values: 8"> 27 > 8 , 9"> 27 > 9 , 1"> 27 > 1 .
Identify the expression with the greatest value: ( 1 + 2 ) 3 .
The expression with the greatest value is: ( 1 + 2 ) 3 .
Explanation
Problem Analysis We are given four expressions and need to find the one with the greatest value. To do this, we will evaluate each expression separately.
Evaluating ( 1 + 2 ) 3 Let's evaluate the first expression: ( 1 + 2 ) 3 . First, we simplify the expression inside the parentheses: 1 + 2 = 3 . Then, we raise the result to the power of 3: 3 3 = 3 × 3 × 3 = 27 .
Evaluating ( 2 × 4 ) 1 Next, we evaluate the second expression: ( 2 × 4 ) 1 . First, we simplify the expression inside the parentheses: 2 × 4 = 8 . Then, we raise the result to the power of 1: 8 1 = 8 .
Evaluating ( 5 − 2 ) 2 Now, we evaluate the third expression: ( 5 − 2 ) 2 . First, we simplify the expression inside the parentheses: 5 − 2 = 3 . Then, we raise the result to the power of 2: 3 2 = 3 × 3 = 9 .
Evaluating ( 15 − 1 ) 0 Finally, we evaluate the fourth expression: ( 15 − 1 ) 0 . First, we simplify the expression inside the parentheses: 15 − 1 = 14 . Then, we raise the result to the power of 0: 1 4 0 = 1 . Remember that any non-zero number raised to the power of 0 is equal to 1.
Comparing the Values We have the following values for the four expressions:
( 1 + 2 ) 3 = 27
( 2 × 4 ) 1 = 8
( 5 − 2 ) 2 = 9
( 15 − 1 ) 0 = 1
Comparing these values, we see that the greatest value is 27, which corresponds to the expression ( 1 + 2 ) 3 .
Examples
Understanding exponents is crucial in many real-world scenarios, such as calculating compound interest. For example, if you invest money in a bank account that offers compound interest, the amount of money you have at the end of each year can be calculated using exponents. Similarly, exponents are used in calculating population growth, radioactive decay, and many other scientific and financial applications. This problem demonstrates the importance of understanding how to evaluate expressions with exponents, which is a fundamental skill in mathematics and its applications.
