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$\frac{7^{16}}{7^{12}}=\frac{7^{-18}}{7^{\square}}$
The missing term in the denominator is $\square$
Answer (1)
Simplify the left side of the equation using the quotient rule: 7 12 7 16 = 7 16 − 12 = 7 4 .
Simplify the right side of the equation using the quotient rule: 7 x 7 − 18 = 7 − 18 − x .
Set the exponents equal to each other: 4 = − 18 − x .
Solve for x : x = − 22 . The missing term in the denominator is 7 − 22 .
− 22
Explanation
Understanding the Problem We are given the equation 7 12 7 16 = 7 x 7 − 18 and we need to find the missing term in the denominator, which is 7 x .
Simplifying the Left Side First, let's simplify the left side of the equation using the quotient rule of exponents, which states that a n a m = a m − n . Applying this rule, we get: 7 12 7 16 = 7 16 − 12 = 7 4
Simplifying the Right Side Now, let's simplify the right side of the equation. We have 7 x 7 − 18 . Using the same quotient rule, we get: 7 x 7 − 18 = 7 − 18 − x
Equating the Exponents Since the left side and the right side of the equation are equal, we can set their exponents equal to each other: 7 4 = 7 − 18 − x Therefore, 4 = − 18 − x
Solving for x Now, we solve for x :
4 = − 18 − x x = − 18 − 4 x = − 22
Finding the Missing Term So, the missing term in the denominator is 7 − 22 .
Examples
Understanding exponents and their properties is crucial in various fields, such as calculating compound interest, where the exponent represents the number of compounding periods. For instance, if you invest 1000 a t anann u a l in t eres t r a t eo f 5 A = P(1 + \frac{r}{n})^{nt} , w h ere P i s t h e p r in c i p a l am o u n t , r i s t h e ann u a l in t eres t r a t e , n i s t h e n u mb ero f t im es t h e in t eres t i sco m p o u n d e d p erye a r , an d t$ is the number of years. Simplifying expressions with exponents helps in determining the growth of investments and other exponential phenomena.
