Simplify: [tex]$2^6-4^0$[/tex]
Answer (1)
Calculate 2 6 = 64 .
Calculate 4 0 = 1 .
Subtract the two results: 64 − 1 = 63 .
The simplified expression is 63 .
Explanation
Understanding the problem We are asked to simplify the expression 2 6 − 4 0 . To do this, we need to evaluate each exponential term separately and then perform the subtraction.
Calculating 2 to the power of 6 First, let's evaluate 2 6 . This means 2 multiplied by itself 6 times: 2 6 = 2 × 2 × 2 × 2 × 2 × 2 = 64
Calculating 4 to the power of 0 Next, let's evaluate 4 0 . Any non-zero number raised to the power of 0 is equal to 1 . Therefore, 4 0 = 1
Performing the subtraction Now, we subtract the result of 4 0 from the result of 2 6 : 2 6 − 4 0 = 64 − 1 = 63
Final result Therefore, the simplified expression is 63 .
Examples
Understanding exponents is crucial in many real-world applications, such as calculating compound interest, modeling population growth, and determining the decay of radioactive materials. For instance, if you invest money with compound interest, the future value of your investment can be calculated using an exponential function. Similarly, exponential functions are used to predict how quickly a population will grow or how long it will take for a radioactive substance to decay to a safe level. These concepts are fundamental in finance, biology, and physics.
