Power Equations

A power equation is an equation of the form $x^{n} = a$ , where $a$ is a fixed number and $n$ is a positive integer. The number of real solutions depends on whether the exponent is odd or even.

Quick Reference

Condition	Solution(s)
$n$ is odd	$x = \sqrt[n]{a}$ (one real solution for any $a$ )
$n$ is even, $a > 0$	$x = \pm \sqrt[n]{a}$ (two real solutions)
$n$ is even, $a = 0$	$x = 0$ (one real solution)
$n$ is even, $a < 0$	No real solution

Consider an equation of the form

x^{n} = a

where $a$ is a fixed number and $n$ is an integer. Then the solutions of the above equation are:

x=\begin{cases} \sqrt[n]{a} & \text{($n$ is odd)}\\ \pm\sqrt[n]{a} & \text{($n$ is even and $a>0$)}\\ \text{no real solution} & \text{($n$ is even and $a<0$)} \end{cases}

Solve $(x - 5)^{4} = 81$ .

Solution

\begin{aligned} x-5 &= \pm\sqrt[4]{81} && \text{(take 4th root of both sides)}\\ x-5 &= \pm\sqrt[4]{3^{4}}\\ x-5 &= \pm3\\ x &= 5\pm3 && \text{(add 5 to both sides)} \end{aligned} Therefore, the solutions are

x = 8

and

x = 2

Solve $(u - 1)^{3} = - 8$ .

Solution

\begin{aligned} (u-1)^3 &= (-2)^3 \\ u-1 &= -2 && \text{(take 3rd root of both sides)}\\ u &= -2+1 = -1. \end{aligned}

Frequently Asked Questions

What is a power equation?

A power equation is any equation of the form

x^{n} = a

, where

n

is a positive integer and

a

is a known constant. Solving it requires taking the

n

th root of both sides, with the number of real solutions depending on whether

n

is odd or even.

Why are there two solutions when the exponent is even?

When

n

is even, both a positive and a negative number raised to the

n

th power give the same positive result. For example,

2^{4} = 16

and

(- 2)^{4} = 16

. So if

x^{n} = a

with

n

even and

a > 0

, both

x = \sqrt[n]{a}

and

x = - \sqrt[n]{a}

are valid solutions.

Why is there no real solution when the exponent is even and

a < 0

Any real number raised to an even power is non-negative. Therefore,

x^{n} \geq 0

for all real

x

when

n

is even, making it impossible for

x^{n}

to equal a negative number.

Why is there exactly one solution when the exponent is odd?

When

n

is odd, the function

x \mapsto x^{n}

is strictly increasing and takes every real value exactly once, including negative values. So for any real

a

, the equation

x^{n} = a

has exactly one real solution:

x = \sqrt[n]{a}

What if

a = 0

a = 0

, then

x^{n} = 0

has only one solution,

x = 0

, regardless of whether

n

is odd or even.

Power Equations

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Power Equations

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