Question

How do you solve \[{2^{{x^2}}} = 32({2^{4x}})\] ?

Answer 1

Hint: Here the question need to find the value of the variable, here the variable is on the power of the number two on both the side of the equation, by equating variables, we can find the value of the variable, here exponential rule of algebra will be used.
Formulae Used:
 \[ \Rightarrow {x^a} \times {x^b} = {x^{a + b}}\] exponential formulae for same variables having some power and are in product form.

Complete step-by-step answer:
Here the given question is \[{2^{{x^2}}} = 32({2^{4x}})\]
Here in order to solve this question first we need to convert the equation in the power of two, on both side of equation then on comparison of powers we can solve further, on solving we get:
First on the right side of the equation we have to solve the number thirty two and put in the power of two, on solving we get:
 \[ \Rightarrow 32 = {2^5}\]
Now solving right hand side of the equation by putting the value of thirty two in terms of power of two, we get:
 \[ \Rightarrow 32({2^{4x}}) = {2^5} \times {2^{4x}} = {2^{5 + 4x}}\,({x^a} \times {x^b} = {x^{a + b}})\]
Now on comparing the powers on the number two on both side of equation, we can get the value of the variable, on solving we get:
 \[
   \Rightarrow {2^{{x^2}}} = {2^{5 + 4x}} \\
   \Rightarrow {x^2} = 5 + 4x \\
   \Rightarrow {x^2} - 4x - 5 = 0 \;
 \]
Here we obtain the equation for the variable,on solving the equation by using mid term splitting rule we get:
 \[
   \Rightarrow {x^2} = 5 + 4x \\
   \Rightarrow {x^2} - 4x - 5 = 0 \\
   \Rightarrow {x^2} - (5 - 1)x - 5 = 0 \\
   \Rightarrow {x^2} - 5x + x - 5 = 0 \\
   \Rightarrow x(x - 5) + 1(x - 5) = 0 \\
   \Rightarrow (x + 1)(x - 5) = 0 \\
   \Rightarrow x = - 1,5 \;
 \]
Here we get the values for our variable.
So, the correct answer is “ x = - 1,5”.

Note: The given question need to be solve by using exponential rule of algebra in which power will be added if same number or variables are in product form, and the power of the values which are on two sides of equation can be compared if the number or variable which is having the power are same.