Solving equations with roots and powers

In general it is hard to find a solution of an equation that involves roots and powers, such as

3x+2x2/3+1=03\sqrt{x}+2x^{2/3}+1=0

An exception is if the equation can be brought into the form

xn=dx^n=d

where dd is a number. For example,

x3/4=5x^{3/4}=5

How do we solve this equation? Here are a couple of methods:

Method 1

x3/4=5∣(a)4x3=54=625∣a3x=6253=8.55‾\begin{array}{lll} x^{3/4} &=& 5 &\quad\vert (\text{\phantom{a}})^4\\ x^3 &=& 5^4=625 &\quad\vert \sqrt[3]{\text{\phantom{a}}}\\ x &=& \sqrt[3]{625}=\underline{8.55} & \end{array}

Methode 2

Instead of taking the third root, we could have raised both sides by 1/31/3:

x3/4=5∣(a)4x3=54=625∣(a)1/3x=6251/3=8.55‾\begin{array}{lll} x^{3/4} &=& 5 &\quad\vert (\text{\phantom{a}})^4\\ x^3 &=& 5^4=625 &\quad\vert (\text{\phantom{a}})^{1/3}\\ x &=& 625^{1/3}=\underline{8.55} & \end{array}

Methode 3

Finally, we could combine the two steps of raising both sides by 44, and then by 1/31/3 by directly raising both sides by 4/34/3:

x3/4=5∣(a)4/3x=54/3=8.55‾\begin{array}{lll} x^{3/4} &=& 5 &\quad\vert (\text{\phantom{a}})^{4/3}\\ x &=& 5^{4/3}=\underline{8.55} & \end{array}
Summary 1

Eine Gleichung der Form

xn/m=dx^{n/m}=d

kann gelöst werden, indem beide Seiten mit dem Umkehrwert m/nm/n potenziert werden:

xn/m=d∣(a)m/nx=dm/n\begin{array}{lll} x^{n/m} &=& d &\quad\vert (\text{\phantom{a}})^{m/n}\\ x &=& d^{m/n} & \end{array}

The same method also works, of course, if nn or mm or both are negative. And note that sometimes we have to do some work in order to bring an equation into the form xn=dx^n=d. Here are some examples.

Example 1

Solve the equations

  1. 5x2/3−4=05x^{2/3} -4 =0

  2. x−1.212=4x^{-1.212}=4

  3. 2x3=8\frac{2}{\sqrt[3]{x}}=8

  4. 3x2/3=12x3x^{2/3} = 12\sqrt{x}

Solution

  1. We have

    5x2/3−4=0∣+4,:5x2/3=0.8∣()3/2 as 12/3=3/2x=0.83/2=0.7155...‾\begin{array}{rll} 5x^{2/3}-4 &=& 0 & \quad\vert +4, :5\\ x^{2/3} &=& 0.8 & \quad\vert ()^{3/2}\, \text{as } \frac{1}{2/3}=3/2\\ x &=&0.8^{3/2}=\underline{0.7155...} & \end{array}

  2. It is

    x−1.212=4∣()−1/1.212 as 1−1.212=−1/1.212x=4−1/1.212=0.318‾\begin{array}{rll} x^{-1.212} &=& 4 & \quad\vert ()^{-1/1.212}\, \text{as } \frac{1}{-1.212}=-1/1.212\\ x &=& 4^{-1/1.212}=\underline{0.318} & \\ \end{array}

  3. It is

    2x3=821x3=82x−1/3=8∣:2x−1/3=4∣()−3 as 1−1/3=−3x=4−3=0.015625‾\begin{array}{rll} \frac{2}{\sqrt[3]{x}} &=& 8 & \\ 2 \frac{1}{\sqrt[3]{x}} &=& 8 & \\ 2 x^{-1/3} &=& 8 & \quad\vert :2\\ x^{-1/3} &=& 4 & \quad\vert ()^{-3}\, \text{as } \frac{1}{-1/3}=-3 \\ x &=& 4^{-3}=\underline{0.015625} & \\ \end{array}

  4. It is

    3x2/3=12x3x2/3=12x1/2∣:3,:x1/2x2/3x1/2=4x1/6=4∣()6 as 11/6=6x=46=4096‾\begin{array}{rll} 3x^{2/3} &=& 12\sqrt{x} & \\ 3x^{2/3} &=& 12x^{1/2} & \quad\vert :3, :x^{1/2}\\ \frac{x^{2/3}}{x^{1/2}} &=& 4 & \\ x^{1/6} &=& 4 & \quad\vert ()^{6}\, \text{as } \frac{1}{1/6}=6\\ x&=& 4^6=\underline{4096} & \\ \end{array}

    Another solution is x=0‾x=\underline{0}, as you can see by inserting x=0x=0 directly into the equation.

We can generalise further by replacing xx with a more complicated expression. Here are two examples:

Example 2

Solve the equation

  1. (3x)1.25=2(3x)^{1.25} = 2

  2. 4⋅x2−13=84\cdot \sqrt[3]{x^2-1}=8

Solution

  1. We have

    (3x)1.25=2∣()1/1.253x=21/1.25=1.741∣:3x=0.58‾\begin{array}{rll} (3x)^{1.25} &=& 2 & \quad\vert ()^{1/1.25}\\ 3x &=& 2^{1/1.25}=1.741 & \quad\vert :3\\ x &=& \underline{0.58} & \end{array}

  2. It is

    4x2−13=8∣:4x2−13=2(x2−1)1/3=2∣()3 as 11/3=3x2−1=23∣+1x2=9∣ax=±3‾\begin{array}{rll} 4\sqrt[3]{x^2-1} &=& 8 & \quad\vert :4\\ \sqrt[3]{x^2-1} &=& 2 & \\ \left(x^2-1\right)^{1/3} &=& 2 & \quad\vert ()^{3}\, \text{as } \frac{1}{1/3}=3\\ x^2-1 &=& 2^3 & \quad\vert+1\\ x^2 &= &9 &\quad\vert \sqrt{\text{\phantom{a}}}\\ x &= &\underline{\pm3} \end{array}
Exercise 1

Solve the following equations

  1. 4x3/4−16=04 x^{3/4}-16=0

  2. 2x0.2=52x^{0.2}=5

  3. 6x2.1=5x6x^{2.1}=5x

  4. 1x34=3\frac{1}{\sqrt[4]{x^3}}=3

  5. 32x1/3=5\frac{3}{2x^{1/3}}=5

  6. 5(x2+4x−1)0.5=105 (x^2+4x-1)^{0.5}=10

  7. 0.5x2/5=x30.5 x^{2/5} =\sqrt[3]{x}

Solution
  1. x=6.349x=6.349
  2. x=97.656x=97.656
  3. x1=0,x2=0.847x_1=0, x_2=0.847
  4. x=0.231x=0.231
  5. x=0.027x=0.027
  6. x1=−5,x2=1x_1=-5, x_2=1
  7. x1=0,x2=32 768x_1=0, x_2=32\,768