Write the expression as an exponent with a base 2 and find its value: [tex]$2^4 \cdot 2$[/tex]
Answer (1)
Rewrite 2 as 2 1 , so the expression is 2 4 c d o t 2 1 .
Apply the exponent rule a m c d o t a n = a m + n to get 2 4 + 1 .
Simplify the exponent to obtain 2 5 .
Calculate the value: 2 5 = b o x e d 32 .
Explanation
Understanding the Problem We are given the expression 2 4 c d o t 2 and asked to write it as an exponent with base 2 and find its value.
Rewriting the Expression First, we can rewrite 2 as 2 1 . So the expression becomes 2 4 c d o t 2 1 .
Applying the Exponent Rule Now, we use the rule of exponents that states a m c d o t a n = a m + n . Applying this rule, we have 2 4 c d o t 2 1 = 2 4 + 1 .
Simplifying the Exponent Simplifying the exponent, we get 2 4 + 1 = 2 5 .
Calculating the Value Finally, we calculate the value of 2 5 . 2 5 = 2 c d o t 2 c d o t 2 c d o t 2 c d o t 2 = 32 .
Examples
Understanding exponents is crucial in many fields, such as computer science, where memory sizes are powers of 2 (e.g., 2^8 = 256 bytes). In finance, compound interest calculations involve exponents. For instance, if you invest $100 at an annual interest rate of 5%, the amount after 3 years is $100 * (1 + 0.05)^3, which involves calculating an exponent. This problem demonstrates a fundamental property of exponents that simplifies calculations and is applicable in various real-world scenarios.
