Simplify. [tex]$\frac{x^{-7}}{x^{-9}}$[/tex]
Write your answer with a positive exponent only.
Answer (2)
Apply the quotient of powers rule: a n a m = a m − n .
Simplify the exponent: − 7 − ( − 9 ) = 2 .
Write the simplified expression: x 2 .
The final answer is x 2 .
Explanation
Understanding the Problem We are given the expression x − 9 x − 7 to simplify. Our goal is to write the answer with a positive exponent only.
Applying the Quotient of Powers Rule To simplify the expression, we will use the quotient of powers rule, which states that a n a m = a m − n . In our case, a = x , m = − 7 , and n = − 9 . Applying this rule, we have: x − 9 x − 7 = x − 7 − ( − 9 )
Simplifying the Exponent Now, we simplify the exponent: − 7 − ( − 9 ) = − 7 + 9 = 2 So, we have: x − 9 x − 7 = x 2
Final Answer Since the exponent is already positive, the simplified expression is x 2 .
Examples
In real life, simplifying expressions with exponents is useful in various fields such as physics and engineering. For example, when dealing with very small or very large numbers in scientific notation, simplifying exponents can make calculations easier and more manageable. Imagine calculating the ratio of the size of an atom ( 1 0 − 10 meters) to the size of a virus ( 1 0 − 7 meters). Simplifying the expression 1 0 − 10 1 0 − 7 gives 1 0 3 , which means the virus is 1000 times larger than the atom.
To simplify x − 9 x − 7 , we use the quotient of powers rule to find x 2 by subtracting the exponents, yielding a positive exponent. The final answer is x 2 .
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