Question

If $ {8^x} = \dfrac{{64}}{{{2^x}}} $ , then the value of x is?

Answer 1

Hint: The problem deals with comparing the powers or indices of two numbers using the basic laws of exponents. Firstly, we cross multiply the terms of the equation. Since the two numbers whose powers are to be equated are not the same, we have to first make the bases of exponents the same before comparing the exponents. For making the bases the same, we use basic exponent rules.

Complete step-by-step answer:
So, we have, $ {8^x} = \dfrac{{64}}{{{2^x}}} $ .
Now, cross multiplying the terms of the equation, we get,
 $ \Rightarrow {8^x} \times {2^x} = 64 $
Now, we know that $ 8 = {2^3} $ . So, we get,
 $ \Rightarrow {\left( {{2^3}} \right)^x} \times {2^x} = 64 $
Now, we know the law of exponent \[{\left( {{a^x}} \right)^y} = {a^{xy}}\]. So, we get,
 $ \Rightarrow {2^{3x}} \times {2^x} = 64 $
Now, we simplify the equation further by using the law of exponents $ {a^x} \times {a^y} = {a^{x + y}} $ .
  $ \Rightarrow {2^{3x + x}} = 64 $
Adding up the powers, we get,
 $ \Rightarrow {2^{4x}} = 64 $
Making use of laws of exponents to make the bases same because exponents can be compared only when the bases are equal.
 $ \Rightarrow {2^{4x}} = {2^6} $
Now we have the same bases on both sides, so now we can equate exponents or powers of both sides of the equation.
Comparing the exponents,
 $ \Rightarrow 4x = 6 $
Dividing both the sides of the equation by $ 4 $ , we get,
 $ \Rightarrow x = \dfrac{6}{4} $
Cancelling the common factors in numerator and denominator, we get,
 $ \Rightarrow x = \dfrac{3}{2} $
So, we get $ x = \dfrac{3}{2} $ on solving the given exponential equation by equating exponents after making the bases the same using laws of exponents.
So, the correct answer is “ $ x = \dfrac{3}{2} $”.

Note: We can also solve the given exponential equation by use of logarithms by taking \[\log \] to the base \[10\] on both sides. Then, solving using simple algebraic rules like transposition, we get the same answer as by the former method. So, exponential equations like the one given in the question can be solved by various methods.