Question

Find the value of x if \[{8^{255}} = {32^x}\].

Answer 1

Hint: This problem is based on the rules of indices and powers. We will first convert both the bases into the same number. As we can see 8 and 32 are both the powers of 2. Once we convert the base into the same number then we can easily equate the powers. And that equation will be solved and will give the value of x.

Complete step by step solution:
Given that, \[{8^{255}} = {32^x}\]
Now as we know 8 can be written as the third power of 2 and 32 can be written as the fifth power of 2.
\[{\left( {{2^3}} \right)^{255}} = {\left( {{2^5}} \right)^x}\]
Now we know that, \[{\left( {{a^m}} \right)^n} = {a^{mn}}\]
\[{2^{3 \times 255}} = {2^{5x}}\]
Now since the bases are the same we can directly equate the powers.
\[3 \times 255 = 5x\]
Now in order to find the value of x,
\[x = \dfrac{{3 \times 255}}{5}\]
On dividing we get,
\[x = 3 \times 51\]
\[x = 153\]
So, the correct answer is “X = 153”.

Note: Note that, though the bases are appearing to be bigger we need not to take them directly. We can convert them into power form. But we can equate the powers only if the bases are the same. Also note that there are different rules applied for solving equations according to the operations taken place between the terms.