What is the value of $x$ in the product of powers 5
$x=\square$
What is the value of $y$ in the product of $p^{-3} \cdot p^{-6}=p^y$ $y=$ $\square$
What is the value of $n$ in the product of $9^2 \cdot 9^{-7} \cdot 9^n=9^{-5}=\frac{1}{9^5}$ ?
$n=\square$
Answer (2)
Find x such that 5 x = 5 , so x = 1 .
Find y such that p − 3 c d o t p − 6 = p y , so y = − 3 + ( − 6 ) = − 9 .
Find n such that 9 2 c d o t 9 − 7 c d o t 9 n = 9 − 5 , so 2 + ( − 7 ) + n = − 5 , which gives n = 0 .
The final answers are x = 1 , y = − 9 , n = 0 .
Explanation
Introduction We are given three separate problems involving the product of powers. Let's solve them one by one.
Solving for x The first problem asks for the value of x in the 'product of powers 5'. This is a bit ambiguous, but we can interpret it as finding x such that 5 x = 5 . Since 5 1 = 5 , we have x = 1 .
Solving for y The second problem asks for the value of y in the product p − 3 c d o t p − 6 = p y . Using the rule for the product of powers, which states that a m c d o t a n = a m + n , we have p − 3 c d o t p − 6 = p − 3 + ( − 6 ) = p − 9 . Therefore, y = − 9 .
Solving for n The third problem asks for the value of n in the product 9 2 c d o t 9 − 7 c d o t 9 n = 9 − 5 . Again, using the product of powers rule, we have 9 2 c d o t 9 − 7 c d o t 9 n = 9 2 + ( − 7 ) + n = 9 − 5 + n . We are given that this equals 9 − 5 . Therefore, − 5 + n = − 5 , so n = 0 .
Final Answer Therefore, the values are x = 1 , y = − 9 , and n = 0 .
Examples
Understanding the product of powers is crucial in many scientific and engineering fields. For example, in computer science, when dealing with memory allocation or data storage sizes, we often encounter powers of 2 (e.g., kilobytes, megabytes, gigabytes). Similarly, in physics, understanding exponential decay or growth involves manipulating powers. Knowing how to simplify expressions with exponents allows for efficient calculations and a deeper understanding of these phenomena.
The values found for the variables are x = 1 , y = − 9 , and n = 0 based on the product of powers rules and equality of exponents.
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