Simplify. Give your answers as positive exponents. [tex]a^5 \cdot a^{-2}[/tex]
A. [tex]a^2[/tex]
B. [tex]\frac{a^2}{a^3}[/tex]
C. [tex]a^3[/tex]
D. [tex]\frac{a^5}{a^2}[/tex]
Answer (2)
Use the property of exponents: a m c d o t a n = a m + n .
Apply the property to the given expression: a 5 c d o t a − 2 = a 5 + ( − 2 ) .
Simplify the exponent: a 5 + ( − 2 ) = a 5 − 2 = a 3 .
The simplified expression is a 3 .
Explanation
Understanding the problem We are given the expression a 5 c d o t a − 2 to simplify. Our goal is to express the answer with positive exponents only.
Applying the exponent property To simplify the expression, we will use the property of exponents that states a m c d o t a n = a m + n , where a is a non-zero number and m and n are integers. In our case, m = 5 and n = − 2 .
Simplifying the expression Applying this property to the given expression, we have: a 5 c d o t a − 2 = a 5 + ( − 2 ) = a 5 − 2 = a 3 So, a 5 c d o t a − 2 = a 3 .
Choosing the correct option Now, we compare the simplified expression a 3 with the given options: A. a 2 B. a 3 a 2 C. a 3 D. a 2 a 5 We see that option C, a 3 , matches our simplified expression.
Final Answer Therefore, the simplified expression is a 3 .
Examples
Understanding exponents is crucial in many fields, such as calculating compound interest. For example, if you invest P dollars at an annual interest rate r compounded n times per year, the amount A you'll have after t years is given by A = P ( 1 + n r ) n t . Simplifying expressions with exponents helps in determining the final amount of your investment.
The expression a 5 ⋅ a − 2 simplifies to a 3 . Therefore, the correct choice is option C: a 3 .
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