Chapter 3.2, Problem 68E

Solution

The simplified expression for the perimeter and area of the given figure.

Perimeter is, $12x^{\frac{1}{3}}$ .

Area is, $6x^{\frac{2}{3}}$ .

Given:

The figure,

  

Concept Used:

• Power of a power property: ${\left(x^a\right)}^b=x^{a\cdot b}$
• Product of power property: $a^m\cdot a^n=a^{m+n}$
• Quotient of power property: $\frac{a^m}{a^n}=a^{m-n}$
• Product property of radicals: $\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}$
• Quotient property of radicals: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
• Sum of like radicals’ property: $p\sqrt[n]{a}+q\sqrt[n]{a}=\left(p+q\right)\sqrt[n]{a}$
• The exponent form of the nth root of a number a is given by $\sqrt[n]{a}=a^{\frac{1}{n}}$
• The perimeter of a triangle is the sum of all its sides and the area is half of the product of the legs of the right triangle.

Calculation:

In order to find the expression for the given rectangle, note that the length of the legs of right-triangle is $3x^{\frac{1}{3}}$ and the width is $4x^{\frac{1}{3}}$ . By using the Pythagorean theorem, the length of the third side is given by

  $\begin{array}{c}\sqrt{{\left(3x^{\frac{1}{3}}\right)}^2+{\left(4x^{\frac{1}{3}}\right)}^2}=\sqrt{9x^{\frac{2}{3}}+16x^{\frac{2}{3}}}\\ =\sqrt{\left(9+16\right)x^{\frac{2}{3}}}\\ =\sqrt{25x^{\frac{2}{3}}}\\ =\sqrt{{\left(5x^{\frac{1}{3}}\right)}^2}\\ =5x^{\frac{1}{3}}\end{array}$

Hence, the perimeter of the triangle is given by

  $\begin{array}{c}3x^{\frac{1}{3}}+4x^{\frac{1}{3}}+5x^{\frac{1}{3}}=\left(3+4+5\right)x^{\frac{1}{3}}\\ =\left(12\right)x^{\frac{1}{3}}\\ =12x^{\frac{1}{3}}\end{array}$

Therefore, the simplified expression for the perimeter is $12x^{\frac{1}{3}}$ .

Since the area of the right-triangle is half of the product of the legs of triangle, it follows

  $\begin{array}{c}\frac{1}{2}\cdot 4x^{\frac{1}{3}}\cdot 3x^{\frac{1}{3}}=\frac{12}{2}\cdot x^{\frac{1}{3}+\frac{1}{3}}\\ =6\cdot x^{\frac{1+1}{3}}\\ =6x^{\frac{2}{3}}\end{array}$

Therefore, the simplified expression for the area is $6x^{\frac{2}{3}}$ .