Represent each of the following expressions as the power of a number a:
[tex]$a \cdot \frac{1}{a^2} \cdot a^5$[/tex]
Answer (2)
Rewrite the expression using exponent rules: a = a 1 and a 2 1 = a − 2 .
Apply the exponent rule a m ⋅ a n = a m + n .
Simplify the expression: a 1 ⋅ a − 2 ⋅ a 5 = a 1 − 2 + 5 .
The final expression is a 4 .
Explanation
Understanding the Problem We are asked to represent the expression a "." a 2 1 "." a 5 as a power of a . This means we want to write the expression in the form a x for some exponent x . To do this, we will use the properties of exponents to simplify the expression.
Rewriting with Exponents First, we rewrite the expression using exponents. Recall that a = a 1 and a 2 1 = a − 2 . Thus, the expression becomes a 1 ⋅ a − 2 ⋅ a 5 .
Simplifying the Expression Next, we use the property of exponents that states a m ⋅ a n = a m + n . Applying this property, we have a 1 ⋅ a − 2 ⋅ a 5 = a 1 + ( − 2 ) + 5 = a 1 − 2 + 5 = a 4 .
Final Answer Therefore, the expression a "." a 2 1 "." a 5 can be represented as a 4 .
Examples
Understanding and manipulating exponents is crucial in many scientific and engineering fields. For example, in physics, the intensity of light decreases with the square of the distance from the source. If the initial intensity is I at a distance of d , then at a distance of 2 d , the intensity becomes I ⋅ ( 2 d ) 2 1 = I ⋅ 4 d 2 1 . This can be simplified using exponent rules to understand how the intensity changes with distance. Similarly, in finance, compound interest calculations rely heavily on exponent rules to determine the growth of investments over time.
The expression a ⋅ a 2 1 ⋅ a 5 simplifies to a 4 by using the rules of exponents. We rewrite terms using negative exponents and then combine them according to exponent rules. The final result is a 4 .
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