Chapter 2.3, Problem 45E

To determine

To find: The real number that is exactly $1$ less than its fourth power.

Answer to Problem 45E

The real numbers that are exactly $1$ less than its fourth power are $x=-0.724$ and $x=1.221$.

Explanation of Solution

Given information: The number is exactly $1$ less than its fourth power.

Calculation:

Assume that $x$ be the real number such that the number is exactly $1$ less than its fourth power. So, the difference between the fourth power of number and number is $1$.

  $x^4-x=1$

Solve the equation $x^4-x=1$ with the help of graph.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter left hand side of the equation after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{X}}\boxed{^}\boxed{4}\boxed{-}\boxed{\text{X}}$ . Now press the $\boxed{\text{ENTER}}$ key and enter right hand side of the equation after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{1}$.

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=\text{X}^4-\text{X}\\ {\text{Y}}_2=1\end{array}$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown below.

  

Figure (1)

As observed from the graph, both the graphs intersect at two points $x=-0.724$ and $x=1.221$.

So, the solution of the equation $x^4-x=1$ are $x=-0.724$ and $x=1.221$.

Therefore, the real number that is exactly $1$ less than its fourth power are $x=-0.724$ and $x=1.221$.