Simplify $4^3 \cdot 4^5$
A. $16 \cdot 15$
B. $4^{15}$
C. $4^8$
D. $16^8$
Answer (2)
Apply the exponent rule a m "." a n = a m + n .
Simplify the expression 4 3 "." 4 5 to 4 3 + 5 .
Calculate the sum of the exponents: 3 + 5 = 8 .
The simplified expression is 4 8 , so the answer is 4 8 .
Explanation
Understanding the Problem We are asked to simplify the expression 4 3 "." 4 5 . This involves using the properties of exponents.
Applying the Exponent Rule The key property we'll use is that when multiplying exponential expressions with the same base, we add the exponents: a m "." a n = a m + n .
Adding the Exponents Applying this property to our expression, we get 4 3 "." 4 5 = 4 3 + 5 .
Simplifying the Expression Simplifying the exponent, we have 4 3 + 5 = 4 8 .
Choosing the Correct Answer Now we compare our simplified expression 4 8 with the given options. Option A is 16"."15 , which is incorrect. Option B is 4 15 , which is also incorrect. Option C is 4 8 , which matches our simplified expression. Option D is 1 6 8 , which is incorrect. Therefore, the correct answer is C.
Examples
Understanding exponent rules is crucial in many fields, such as computer science when dealing with memory sizes (e.g., kilobytes, megabytes, gigabytes) or in physics when calculating quantities that scale exponentially, like radioactive decay. For instance, if a computer's memory doubles every year, we can use exponents to predict its memory size after a certain number of years. Similarly, in finance, compound interest calculations rely heavily on exponential growth, helping us understand how investments increase over time.
To simplify 4 3 ⋅ 4 5 , we add the exponents to get 4 8 . The correct answer from the options provided is 4 8 . Therefore, the answer is C.
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