Question

If \[25025 = {p_1}^{{x_1}}.{p_2}^{{x_2}}.{p_3}^{{x_3}}.{p_4}^{{x_4}}\] , find the value of \[{p_1}\], \[{p_2}\],\[{p_3}\], \[{p_4}\] and \[{x_1}\],\[\;{x_2}\], \[{x_3}\], \[{x_4}\]

Answer 1

Hint: First, we need to analyze the given information so that we are able to obtain the required answer. It is given that \[25025 = {p_1}^{{x_1}}.{p_2}^{{x_2}}.{p_3}^{{x_3}}.{p_4}^{{x_4}}\].
We are asked to calculate the value of the unknown variables.
We need to factorize \[25025\]
After factoring, we have to compare the resultant equation with the given equation.
So by directly factoring \[25025\], we can easily solve this problem.

Complete step by step answer:
It is given that \[25025 = {p_1}^{{x_1}}.{p_2}^{{x_2}}.{p_3}^{{x_3}}.{p_4}^{{x_4}}\].
Now, we need to factorize\[25025\] as follows.
 \[\begin{array}{*{20}{c}}
  {5\left| \!{\underline {\,
  {25025} \,}} \right. } \\
  {5\left| \!{\underline {\,
  {5005} \,}} \right. } \\
  {11\left| \!{\underline {\,
  {1001} \,}} \right. } \\
  {7\left| \!{\underline {\,
  {91} \,}} \right. } \\
  {13\left| \!{\underline {\,
  {13} \,}} \right. } \\
  {1\left| \!{\underline {\,
  1 \,}} \right. } \\
  {\,\,}
\end{array}\]
Hence, \[25025 = 5 \times 5 \times 7 \times 11 \times 13\;\]
\[ = {5^2} \times {7^1} \times {11^1} \times {13^1}\;\]...............\[(1)\]
It is given that \[25025 = {p_1}^{{x_1}}.{p_2}^{{x_2}}.{p_3}^{{x_3}}.{p_4}^{{x_4}}\]……………...\[\left( 2 \right)\]
After factoring, we have to compare the resultant equation with the given equation.
Here, we can notice that both the first equation and the second equation 2 are in the same format. So we can compare \[(1)\]and\[\left( 2 \right)\].
\[{p_1}^{{x_1}}.{p_2}^{{x_2}}.{p_3}^{{x_3}}.{p_4}^{{x_4}} = {5^2} \times {7^1} \times {11^1} \times {13^1}\;\]
Now, we need to understand that the base numbers are to be compared and the exponent or powers are to be compared like \[{p_1}^{{x_1}} = {5^2}\] . Here the base numbers are \[{p_1} = 5\;\] and the exponents \[{x_1} = 2\] . Similarly, all the factors are to be compared.
\[{p_1}^{{x_1}} = {5^2}\]
\[{p_2}^{{x_2}} = {7^1}\;\]
\[{p_3}^{{x_3}} = {11^1}\]
\[{p_4}^{{x_4}} = {13^1}\;\]
From the above equations, we shall compare the base numbers and their exponents.
Hence, we get,
\[{p_1}^{{x_1}} = {5^2} \Rightarrow {p_1} = 5\;,{x_1} = 2\]
\[{p_2}^{{x_2}} = {7^2} \Rightarrow {p_2} = 7\;,{x_2} = 1\]
\[{p_3}^{{x_3}} = {11^1} \Rightarrow {p_3} = 11\;,{x_3} = 1\]
\[{p_4}^{{x_4}} = {13^1} \Rightarrow {p_4} = 13\;,{x_4} = 1\]
Therefore, the required answer is \[{p_1} = 5\], \[{p_2} = 7\],\[{p_3} = 11\], \[{p_4} = 13\] and \[{x_1} = 2\],\[\;{x_2} = 1\], \[{x_3} = 1\], \[{x_4} = 1\]

Note: In this question, we need to factorize $25025$
After factoring, we have to compare the resultant equation with the given equation.
Here, we learn that the base numbers are to be compared and the exponent or powers are to be compared like \[{p_1}^{{x_1}} = {5^2}\] . Here the base numbers are \[{p_1} = 5\;\] and the exponents \[{x_1} = 2\] . Similarly, all the factors are to be compared.