Simplify the expression below.
$\left(x^6\right)\left(x^2\right)$
A. $x^4$
B. $x^6$
C. $x^8$
D. $x^{12}$
Answer (1)
Use the exponent rule x a "." x b = x a + b .
Apply the rule to the expression: ( x 6 ) ( x 2 ) = x 6 + 2 .
Add the exponents: 6 + 2 = 8 .
The simplified expression is x 8 .
Explanation
Understanding the Problem We are asked to simplify the expression ( x 6 ) ( x 2 ) . This involves using the properties of exponents to combine the terms.
Applying the Exponent Rule To simplify the expression, we use the rule that when multiplying terms with the same base, we add the exponents: x a ⋅ x b = x a + b .
Calculation Applying this rule to our expression, we have ( x 6 ) ( x 2 ) = x 6 + 2 .
Simplifying the Exponent Now, we simply add the exponents: 6 + 2 = 8 . Therefore, the simplified expression is x 8 .
Final Answer Comparing our result to the given options, we see that the correct answer is C. x 8 .
Examples
Understanding exponents is crucial in many fields, such as computer science when dealing with memory sizes (e.g., kilobytes, megabytes, gigabytes, which are powers of 2) or in physics when calculating quantities that scale exponentially, like radioactive decay or compound interest. For instance, if a population of bacteria doubles every hour, the growth can be modeled using exponents, helping predict the population size at any given time. This concept is also fundamental in understanding algorithms and data structures, where efficiency is often expressed in terms of exponential or logarithmic functions.
