Write the expression as an exponent with a base 2 and find its value: [tex]$2^6 \cdot 4$[/tex]
Answer (2)
Rewrite 4 as a power of 2: 4 = 2 2 .
Apply the exponent property a m c d o t a n = a m + n : 2 6 c d o t 2 2 = 2 6 + 2 .
Simplify the exponent: 2 6 + 2 = 2 8 .
Calculate the value: 2 8 = 256 . The expression 2 6 c d o t 4 as an exponent with base 2 is 2 8 , and its value is 256 .
Explanation
Understanding the Problem We are given the expression 2 6 c d o t 4 and asked to write it as an exponent with base 2 and find its value.
Rewriting 4 as a Power of 2 First, we need to express 4 as a power of 2. We know that 4 = 2 c d o t 2 = 2 2 . So we can rewrite the expression as 2 6 c d o t 2 2 .
Applying the Exponent Property Now we can 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 get 2 6 c d o t 2 2 = 2 6 + 2 .
Simplifying the Exponent Simplifying the exponent, we have 2 6 + 2 = 2 8 .
Calculating the Value Finally, we calculate the value of 2 8 . 2 8 = 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 = 256 .
Examples
Exponents are useful in many real-world situations, such as calculating compound interest, modeling population growth, and understanding exponential decay in radioactive materials. For example, if a bacteria population doubles every hour, the population after t hours can be modeled as P ( t ) = P 0 c d o t 2 t , where P 0 is the initial population. Understanding how to manipulate exponents allows us to predict and analyze these phenomena effectively.
The expression 2 6 ⋅ 4 can be rewritten as 2 8 by expressing 4 as 2 2 . The value of 2 8 is 256. Therefore, the final result is 2 8 = 256 .
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