Question

If $ {{3}^{x}}=500 $ then the value of $ {{3}^{x-2}} $ is
\[\begin{align}
  & A.\dfrac{100}{9} \\
 & B.\dfrac{1000}{9} \\
 & C.\dfrac{500}{9} \\
 & D.\dfrac{500}{3} \\
\end{align}\]

Answer 1

Hint: In this question, we are given the value of $ {{3}^{x}} $ and we need to find the value of $ {{3}^{x-2}} $ . For this, we will use the properties of exponent and simplify our answer to get the required value. We will use the property $ {{a}^{m-n}}=\dfrac{{{a}^{m}}}{{{a}^{n}}} $ to evaluate our answer.

Complete step by step answer:
Here we are given the value of $ {{3}^{x}} $ as 500. We need to find the value of $ {{3}^{x-2}} $ . As we know from the property of exponents that $ {{a}^{m-n}}=\dfrac{{{a}^{m}}}{{{a}^{n}}} $ . So here we have an expression as $ {{3}^{x-2}} $ . Taking 'a' as 3, m as x and n as 2. We get $ {{3}^{x-2}}=\dfrac{{{3}^{x}}}{{{3}^{2}}} $ .
Now we know that the value of $ {{3}^{x}} $ is 500, so putting in the value we get $ {{3}^{x-2}}=\dfrac{500}{{{3}^{2}}} $ .
We know that $ {{3}^{2}} $ can be written as $ 3\times 3 $ and the value of $ 3\times 3 $ is 9, therefore $ {{3}^{2}}=9 $ . Putting this value in the above equation we get, $ {{3}^{x-2}}=\dfrac{500}{9} $ which is our required answer.
Hence the value of $ {{3}^{x-2}} $ is $ \dfrac{500}{9} $ .
Hence option C is the correct answer.

Note:
 Students should take care of the sign in the power of exponent as the formula changes with sign. For $ {{a}^{m-n}} $ we have the value as $ \dfrac{{{a}^{m}}}{{{a}^{n}}} $ but for $ {{a}^{m+n}} $ , we have the value as $ {{a}^{m}}\cdot {{a}^{n}} $ . Make sure that the base remains the same. Note that $ {{x}^{a}} $ means to multiply x by itself a times. Students should not try to find the value of x first from the given expression and then put it to the given expression for finding the solution. If the options are not given, try to find the final answer in either mixed fraction or in decimal, do not leave the answer in an improper fraction.