Simplify the expression: [tex]$k^n \cdot k$[/tex]
Answer (2)
Recognize that k is equivalent to k 1 .
Apply the exponent rule $a^m
times a^n = a^{m+n}$.
Add the exponents: n + 1 .
The simplified expression is k n + 1 .
Explanation
Understanding the Problem We are given the expression $k^n
times k , w h ere k i s a v a r iab l e an d n$ is an exponent. We want to simplify this expression using the properties of exponents.
Recalling the Properties of Exponents Recall the property of exponents: $a^m
times a^n = a^{m+n}$. This property states that when multiplying two exponential expressions with the same base, we add the exponents.
Applying the Property Apply this property to the given expression: $k^n
times k = k^n
times k^1 = k^{n+1} . Here , w ereco g ni ze t ha t k i s t h es am e a s k^1 , so w ec ana dd t h ee x p o n e n t s n$ and 1 .
Final Answer The simplified expression is k n + 1 .
Examples
Imagine you're calculating the area of a rectangle where one side is k n units and the other side is k units. The total area would be $k^n
times k = k^{n+1}$ square units. This kind of simplification is useful in various fields like physics, engineering, and computer science, where exponential relationships are common. For example, in compound interest calculations, simplifying expressions with exponents can help determine the total amount accumulated over time. Understanding and applying exponent rules makes complex calculations more manageable and provides insights into the relationships between variables.
The expression k n ⋅ k can be simplified by recognizing that k = k 1 , and applying the property of exponents. This results in k n ⋅ k = k n + 1 . Therefore, the simplified expression is k n + 1 .
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