Solving Equations with Exponents

Solving Equations with Exponents.

Consider these two equations:

Equation 1: x2 = 4 and Equation 2: x3 = 27

Equation 1 has two solutions: 2 and -2 since 22 = 4 and (-2)2 = 4.

Equation 2 only has one solution: x = 3.

Whenever an equation contains all even exponents, you should consider both the positive and negative solutions. If the exponent is an odd power, there is only one solution.

Solving Equations with Exponents: xm=k

If m is even: x = ±m k

If m is odd: x = m k

For equations which include roots other than the square root, you want to remove the roots by (1) isolating the root term on one side of the equation, and (2) raising both sides of the equation to the appropriate power.

Example 1. Solve (x² + 6x)1/4 = 2

Solution

Recall that a fractional exponent is actually a root: am/n = (n a ) m

Remove the 4th root by raising each side of the equation to the 4th power.

[(x² + 6x)1/4]4 = 24

Simplify each side of the equation.

x² + 6x = 16

Set the equation equal to zero.

x² + 6x − 16 = 0

Factor the left side of the equation.

(x + 8)(x − 2) = 0

Set factors equal to zero and solve.

0 = x + 8 or 0 = x − 2

x = − 8 or x = 2

Our possible solutions are x = − 8 and x = 2. Both of these solutions need to be checked using the original equation.

Check x = − 8:

[(− 8)² + 6(− 8)]1/4 = 2

[64 − 48]1/4 = 2

[16]1/4 = 2

4√ 16 = 2

2 = 2 is a true statement. Therefore x = − 8 is a solution.

Check x = 2:

[(2)² + 6(2)]1/4 = 2

[4 + 12]1/4 = 2

[16]1/4 = 2

4√ 16 = 2

2 = 2 is a true statement. Therefore x = 2 is a solution.

The solutions to the equation,(x² + 6x)1/4 = 2, are x = − 8 and x = 2.

Example 2 : Solve for w: 5w2/3 + 3 = 23

Solution.

Isolate the w-term on the left side of the equation. Subtract 3 from each side of the equation.

5w2/3 = 23 − 3

5w2/3 = 20

Divide each side of the equation by 5.

w2/3 = 20 ÷ 5

w2/3 = 4

Isolate the w by raising both sides of the equation to the 3/2 power. Since the numerator of the exponent is even, there will be two answers.

w = ±43/2 = ± (√ 4 )3

w = ±23 = ±8

The two answers to the equation, 5w2/3 + 3 = 23, are 8 and -8.

Solve x\frac{2}{5} = 4

Solve x\frac{3}{2} = 27