Chapter 3, Problem 7VE

Solution

The step to solve the equation $x^{\frac{2}{3}}=5$ in just one step.

  $\text{Radical function}$

Given:

The equation $x^{\frac{2}{3}}=5$ .

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 $\sqrt[n]{a}=a^{\frac{1}{n}}$

Calculation:

The student start solving the equation $x^{\frac{2}{3}}=5$ by cubing each side to remove the 3 from the denominator in the exponent on left side. The next step would be to take the square root both sides to solve for x as shown below

  $\begin{array}{c}{x}^{\frac{2}{3}}=5\\ {\left(x^{\frac{2}{3}}\right)}^3={\left(5\right)}^3\\ x^{\frac{2}{3}\cdot 3}={\left(5\right)}^3\\ x^2=5^3\\ {\left(x^2\right)}^{\frac{1}{2}}={\left(5^3\right)}^{\frac{1}{2}}\\ x^{2\cdot \frac{1}{2}}=5^{3\cdot \frac{1}{2}}\\ x=5^{\frac{3}{2}}\end{array}$

In order to solve the given equation in just one step, student should have raised the exponent both sides by the reciprocal of $\frac{2}{3}$ , that is, should have raised the exponents both side by $\frac{3}{2}$ as shown below

  $\begin{array}{c}{x}^{\frac{2}{3}}=5\\ {\left(x^{\frac{2}{3}}\right)}^{\frac{3}{2}}={\left(5\right)}^{\frac{3}{2}}\\ x^{\frac{2}{3}\cdot \frac{3}{2}}=5^{\frac{3}{2}}\\ x=5^{\frac{3}{2}}\end{array}$