Multiply: [tex]a^{-5}(a^3)(a^{-2})[/tex]
A. [tex]\frac{1}{a^5}[/tex]
B. [tex]a^4[/tex]
C. [tex]a^{30}[/tex]
D. [tex]\frac{1}{a^4}[/tex]
Answer (2)
We have to multiply the expression a − 5 ( a 3 ) ( a − 2 ) .
Using the property of exponents, we add the exponents: a − 5 + 3 + ( − 2 ) .
Simplifying the exponent, we get a − 4 .
Rewriting with a positive exponent, we have a 4 1 .
The final answer is a 4 1 .
Explanation
Understanding the problem We are asked to multiply the expression a − 5 ( a 3 ) ( a − 2 ) . To do this, we need to use the properties of exponents. Specifically, when multiplying terms with the same base, we add their exponents.
Applying the exponent rule Using the property of exponents, we have a − 5 ( a 3 ) ( a − 2 ) = a − 5 + 3 + ( − 2 )
Simplifying the exponent Now, we simplify the exponent: − 5 + 3 + ( − 2 ) = − 5 + 3 − 2 = − 4 So, the expression simplifies to a − 4 .
Rewriting with positive exponent We can rewrite a − 4 using the property a − n = a n 1 . Therefore, a − 4 = a 4 1
Choosing the correct answer Comparing our simplified expression a 4 1 with the given options, we see that it matches option D.
Examples
Understanding exponents 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 or compound interest. For instance, if a computer's memory doubles every year, the memory size can be expressed as an exponential function of time. Similarly, the intensity of light decreases exponentially as it passes through a medium, which is vital in optics and environmental science.
To multiply the expression a − 5 ( a 3 ) ( a − 2 ) , we add the exponents to get a − 4 . This simplifies to a 4 1 , which corresponds to option D. Therefore, the answer is a 4 1 .
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