Write the expression as an exponent with a base 2 and find its value: [tex]$8 \cdot 2^7$[/tex]
Answer (1)
Rewrite 8 as 2 3 .
Substitute 8 = 2 3 into the expression: 2 3 c d o t 2 7 .
Use the property of exponents: a m c d o t a n = a m + n .
Apply the property to simplify the expression to 2 3 + 7 = 2 10 and calculate the value of 2 10 = 1024 . The final answer is 1024 .
Explanation
Understanding the Problem We are given the expression 8 c d o t 2 7 and asked to write it as an exponent with base 2 and find its value.
Expressing 8 as a Power of 2 First, we express 8 as a power of 2. Since 8 = 2 c d o t 2 c d o t 2 = 2 3 , we can rewrite the expression as 2 3 c d o t 2 7 .
Applying the Exponent Property Next, we use the property of exponents that states a m c d o t a n = a m + n . Applying this property to our expression, we have 2 3 c d o t 2 7 = 2 3 + 7 = 2 10 .
Calculating the Value Finally, we calculate the value of 2 10 . 2 10 = 2 c d o t 2 c d o t 2 c d o t 2 c d o t 2 c d o t 2 c d o t 2 c d o t 2 c d o t 2 c d o t 2 = 1024 .
Examples
Understanding exponents is crucial in many fields, such as computer science where memory sizes are powers of 2 (e.g., kilobytes, megabytes, gigabytes). For instance, if you're calculating the storage capacity of a memory chip and it's given as 2 12 bytes, you can quickly determine it's 4096 bytes. Similarly, in finance, compound interest calculations often involve exponents, helping you determine the future value of an investment.
