Question

A number to the $8th$ power is divided by the same number to the $4th$ power is $16$. What is the number?
A) $2$
B) $6$
C) \[4\]
D) $8$

Answer 1

Hint: $x$ to the power $a$ is written as${x^a}$
Let any number and write the given condition in the mathematical form using the above expression and solve the expression to get the number.

Complete step-by-step answer:
We are given that a number to the $8th$ power is divided by the same number to the $4th$ power is $16$.
First, we assume the number.
Let the number be $x$.
Now we convert our given statement into the mathematical form in order to evaluate the number.
We know that $x$ to the power $a$is written as${x^a}$
Therefore, $x$ to the power $8$is written as${x^8}$
Similarly, $x$ to the power $4$is written as${x^4}$
Now we are given that the first number is divided by the second number and the result is $16$.
Write the mathematical expression.
$\dfrac{{{x^8}}}{{{x^4}}} = 16$
Now we solve our expression.
We solve it by applying the property of exponents.
Here we use the quotient property of the exponents.
The quotient property of exponents states that when you divide powers with the same base you just have to subtract the exponents.
Mathematically it can be written as,
$\dfrac{{{x^a}}}{{{x^b}}} = {x^{a - b}}$
Apply the property in our equation.
\[
  {x^{8 - 4}} = 16 \\
  {x^4} = 16 \\
 \]
Rewrite the right-hand side of the equation in the power of $4$.
${x^4} = {2^4}$
Compare both the sides to evaluate the value of $x$.
$x = 2$
Hence, the required number is $2$.
Therefore, option (A) is correct.
Note: We can solve our equation by an alternate method which is shown below:
$\dfrac{{{x^8}}}{{{x^4}}} = 16$
Cross multiply and solve the equation.
\[
  {x^8} = 16{x^4} \\
  {x^8} - 16{x^4} = 0 \\
 \]
Factor out the common factor ${x^4}$.
\[
  {x^4}({x^4} - 16) = 0 \\
  x \ne 0 \\
  \therefore {x^4} - 16 = 0 \\
  {x^4} = 16 \\
  {x^4} = {2^4} \\
  x = 2 \\
 \]
Hence, the required number is .