Question

How do you solve the equation ${{4}^{12-3x}}=64$

Answer 1

Hint: Now to solve the given equation we will first find the factors of 64. Now with the help of factors we will write 64 in powers of 4. Now we know that if we have ${{a}^{m}}={{a}^{n}}$ then m = n. hence using this in the obtained equation we will get a linear equation in x. Now we will separate the variables and the constants and then divide the equation with the coefficient of x. hence we will get the value of x.

Complete step-by-step solution:
Now first let us consider the given equation ${{4}^{12-3x}}=64$
First we will factorize 64 and try to write it in powers of 4.
Now we know that 64 = 2 × 2 × 2 × 2 × 2 × 2
Hence we can say that 64 = 4 × 4 × 4.
Now we have that $64={{4}^{3}}$ hence if we write 64 as ${{4}^{3}}$ in the given equation we get,
$\Rightarrow {{4}^{12-3x}}={{4}^{3}}$
Now we know that if ${{a}^{m}}={{a}^{n}}$ then m = n.
Hence using this we get,
$\Rightarrow 12-3x=3$
Now we have a linear equation in x. To solve this equation we will separate the variables and constants. Hence we get,
$\begin{align}
  & \Rightarrow 12-3=3x \\
 & \Rightarrow 9=3x \\
\end{align}$
Now dividing the whole equation by 3 we get,
$\Rightarrow x=3$
Hence the solution of the given equation is x = 3.

Note: Now note that when we have an equation in exponent form always try to convert it into linear equation by taking factors of the equation. Once we have a linear equation we can easily solve the equation to find the value of x.