Chapter 3.1, Problem 65PS

Solution

(a)

To explain how the given graph justifies the results in the Key Concept box when n is a positive even integer.

  $25.44\text{ cm}$

Given:

The graph of $y=x^n$ for any positive even integer n .

  

And the Key Concept box,

  

Concept Used:

• The even inter root of a number is defined for positive values only that are greater than or equal to 0.
• The odd integer root of all real numbers is defined.
• Power of a power property: ${\left(x^a\right)}^b=x^{a\cdot b}$
• The exponent form of the nth root of a number a is given by $\sqrt[n]{a}=a^{\frac{1}{n}}$

Calculation:

From the given graph, note that the line $y=a\text{ for }a<0$ does not intersect the graph of $y=x^n$ at any point, so there is no solution when the value of a is negative.

Also, note that the line $y=0$ touches the graph of $y=x^n$ at one point only, so there is only one solution when $a=0$ .

And, the line $y=a\text{ for }a>0$ intersects the graph of $y=x^n$ at two different points, so there are two solutions when the value of a is positive.

(b)

To draw a graph that justifies the results in the Key concept box when n is a positive odd integer.

  

Given:

The Key Concept box,

  

Concept Used:

• The even inter root of a number is defined for positive values only that are greater than or equal to 0.
• The odd integer root of all real numbers is defined.
• Power of a power property: ${\left(x^a\right)}^b=x^{a\cdot b}$
• The exponent form of the nth root of a number a is given by $\sqrt[n]{a}=a^{\frac{1}{n}}$

Calculation:

The graph that justifies the results in the Key Concept box when n is a positive odd integer is shown below