Question

Solve the following equation : \[{2^{{x^2}}}:{2^{2x}} = 8:1\].

Answer 1

Hint: We have to find the value of \[x\] from the given expression \[{2^{{x^2}}}:{2^{2x}} = 8:1\] . We solve this using the concept of exponents rules and properties . We will first take the left hand side of the given expression and convert the expression in the simplest terms possible by changing the terms of the ratios in the form of fractional terms and then changing each term in terms of powers of \[2\] and after simplifying the terms of the left hand side we would be comparing the both sides and hence we can solve the expression for the value of \[x\] .

Complete step-by-step solution:
Given :
\[{2^{{x^2}}}:{2^{2x}} = 8:1\]
We know that the terms of the ratios of two numbers can be written as :
\[a:b = \dfrac{a}{b}\]
Using this we can simplify the given expression as :
\[\dfrac{{{2^{{x^2}}}}}{{{2^{2x}}}} = \dfrac{8}{1}\]
Now , we also know that 8 can be written in powers of \[2\] as :
\[8 = {2^3}\]
Putting these values in the expression , we get
\[\dfrac{{{2^{{x^2}}}}}{{{2^{2x}}}} = \dfrac{{{2^3}}}{1}\]
We know that the exponents rule of zero rule is as given below :
\[{b^0} = 1\]
Using the rule of exponents , we get the expression as :
\[\dfrac{{{2^{{x^2}}}}}{{{2^{2x}}}} = \dfrac{{{2^3}}}{{{2^0}}}\]
We also know that the exponents rule of quotient is as given below :
\[\dfrac{{{a^n}}}{{{a^m}}} = {a^{n - m}}\]
Using the rule of exponents , we get the expression as :
\[{2^{{x^2} - 2x}} = {2^{3 - 0}}\]
We also know that the exponent's rule of bases states that when the bases of two variables are the same then their powers are equal .
i.e. if \[{a^n} = {a^m}\]
then \[n = m\]
As both the left hand side and the right hand side of the expression have the same base element I.e. \[2\] .
Hence , Using the rule of exponents , we get the expression for the value of \[x\] as :
\[{x^2} - 2x = 3\]
\[\Rightarrow {x^2} - 2x - 3 = 0\]
On simplifying the expression by splitting the roots , we get
\[{x^2} - 3x + x - 3 = 0\]
\[\Rightarrow x\left( {x - 3} \right) + 1\left( {x - 3} \right) = 0\]
On simplifying further , we get
\[\left( {x + 1} \right) \times \left( {x - 3} \right) = 0\]
\[\Rightarrow x = - 1\] or \[x = 3\]
Hence , the value of \[x\] in the given expression \[{2^{{x^2}}}:{2^{2x}} = 8:1\] are \[ - 1\] and \[3\].

Note: After solving the question we can verify that we have calculated the correct value of \[x\] by putting the obtained value of \[x\] back into the expression and if we obtain both the left hand side and the right hand side equal then we have calculated the correct value .
The various rules and properties for the exponents are as given below :
Product rule : \[{a^n} \times {b^n} = {(a \times b)^n}\]
Quotient rule : \[\dfrac{{{a^n}}}{{{a^m}}} = {a^{n - m}}\]
Zero rule : \[{b^0} = 1\]
One rule : \[{b^1} = b\]