2.7: Solving with rational exponents
- Last updated
- Jan 12, 2023
- Save as PDF
- Page ID
- 121859
( \newcommand{\kernel}{\mathrm{null}\,}\)
In this section, we take solving with radicals one step further and apply radicals and rational exponents to solve equations with exponents. Since radicals have some restrictions on the radicand, we will also have some restrictions here when applying a rational exponents in the solving process.
The Odd Root Property
Let's start with the case that we take an odd root of an equation.
The Odd Root Property
If xn=p, where n is odd, then √x=n√p. Note, the radicand can be any real number, i.e., p is any number in (−∞,∞).
Example 10.7.1
Solve: x5=32
Solution
We can easily apply the odd root property to solve for x.
x5=32Apply odd root property5√x5=5√32Simplifyx=2Solution
Example 10.7.2
Solve: 4r3−2=106
Solution
We can easily apply the odd root property to solve for r.
4r3−2=106Isolate the variable term4r3=108Isolate r3r3=27Apply odd root property3√r3=3√27Simplifyr=3Solution
The Even Root Property
With even roots, we have the restriction on the radicand where the radicand is required to be non-negative here. We discussed this in the previous section, e.g., √−4 is not a real number. We continue this restriction when taking even roots of an equation.
The Even Root Property
If xn=p, where n is even, then x=n√p orx=−n√p or we can write x=±n√p. Note, the radicand can be any real non-negative number, i.e., p≥0.
Example 10.7.3
Solve: x4=16
Solution
We can easily apply the even root property to solve for x.
x4=16Apply even root property4√x4=4√16Simplify|x|=±2x=±2Solution
Notice, it wasn’t given that x≥0. Hence, we cannot assume it is, so we put absolute value around x. Once we verify the solution(s), then we can remove the absolute value around x.
Note
In 1545, French mathematician Gerolamo Cardano published his book The Great Art, or the Rules of Algebra, which included the solution to an equation with a fourth power, but it was considered absurd by many to take a quantity to the fourth power because there are only three dimensions!
Example 10.7.4
Solve: (2x+4)2=36 Find and verify all solutions that satisfy the equation.
Solution
We can easily apply the even root property to solve for x.
(2x+4)2=36Apply even root property√(2x+4)2=±√36Simplify √362x+4=±6Rewrite into two equations2x+4=6or2x+4=−6Isolate the variable term in each equation2x=2or2x=−10Solve each equationx=1orx=−5Solutions
We can always verify the solutions by substituting back in 1, −5 into the original equation:
(2x+4)2=36Plug-n-chug x=1(2(1)+4)2?=36Simplify each side(2+4)2?=3662?=3636=36✓True
Let's try the next solution x=−5:
(2x+4)2=36Plug-n-chug x=−5(2(−5)+4)2?=36Simplify each side(−10+4)2?=36(−6)2?=3636=36✓True
Thus, 1, (-5) are, in fact, solutions to the original equation.
Example 10.7.5
Solve: (6x−9)2=45 Find and verify all solutions that satisfy the equation.
Solution
(6x−9)2=45Apply even root property√(6x−9)2=±√45Simplify √456x−9=±3√5Isolate the variable term6x=9±3√5Divide both sides by 6x=9±3√56Factor a GCF from numeratorx=393±√5)62Simplifyx=3±√52Solution
Notice, we didn’t split the equation into two different equations and solve. Since √45 is an irrational number, we can leave the ± and solve as usual. We leave verifying the solutions to the student.
Example 10.7.6
Solve: 256w8+40=41
Solution
We have to isolate the variable term first, then we can apply the even root property.
256w8+40=41Isolate the variable term.256w8=1Divide each side by 256w8=1256Apply even root property8√w8=±8√1256Simplify the radicals|w|=±12w=±12Solution
Notice, it wasn't given that w≥0. Hence, we cannot assume it is and we put absolute value around w. Once we verify the solution(s), then we can remove the absolute value around w.
Solving Equations with Rational Exponents
When exponents are fractions, we convert the rational exponent into a radical expression to solve. Recall, amn=(n√a)m. Then we clear the exponent by applying either the even or odd root property and solve as usual.
Steps for solving equations with rational exponents
Given an equation with rational exponents, we can follow the following steps to solve.
Step 1. Rewrite any rational exponents as radicals.
Step 2. Apply the odd or even root property. Recall, even roots require the radicand to be positive unless otherwise noted.
Step 3. Raise each side to the power of the root.
Step 4. Solve. Verify the solutions, especially when there is an even root.
Example 10.7.7
Solve: (4x+1)25=9. Assume all variables are positive.
Solution
We follow the steps in order to solve the equation with a rational exponent.
Step 1. Rewrite any rational exponents as radicals. (4x+1)25=9(5√4x+1)2=9
Step 2. Apply the odd or even root property. Recall, even roots require the radicand to be positive unless otherwise noted.
Since we are taking the square root, which is even, then we apply the even root property: (5√4x+1)2=95√4x+1=±√95√4x+1=±3
Step 3. Raise each side to the power of the root.
Since the root is 5, then we can raise each side to the fifth power: 5√4x+1=±3(5√4x+1)5=(±3)54x+1=±243
Step 4. Solve. Verify the solutions, especially when there is an even root. 4x+1=243or4x+1=−2434x=242or4x=−244x=2424orx=−61x=1212orx=−61
Thus, the solutions to the equation are 1212,−61.
Example 10.7.8
Solve: (3x−2)34=64
Solution
We follow the steps in order to solve the equation with a rational exponent.
Step 1. Rewrite any rational exponents as radicals. (3x−2)34=64(4√3x−2)3=64
Step 2. Apply the odd or even root property. Recall, even roots require the radicand to be positive unless otherwise noted.
Since we are taking the cube root, which is odd, then we apply the odd root property: (4√3x−2)3=644√3x−2=3√644√3x−2=4
Step 3. Raise each side to the power of the root.
Since the root is 4, then we can raise each side to the fourth power: 4√3x−2=4(4√3x−2)4=443x−2=256
Step 4. Solve. Verify the solutions, especially when there is an even root. 3x−2=2563x=258x=86
Thus, the solution is 86.
When solving equations with rational exponents, it is very helpful to convert the equations into their radical form so we can see which property we need to use and to identify whether we need to verify the solutions due to an even root in the original equation.
Solving with Rational Exponents Homework
Solve.
Exercise 10.7.1
x2=75
Exercise 10.7.2
x2+5=13
Exercise 10.7.3
3x2+1=73
Exercise 10.7.4
(x+2)5=−243
Exercise 10.7.5
(2x+5)3−6=21
Exercise 10.7.6
(x−1)23=4
Exercise 10.7.7
(2−x)32=27
Exercise 10.7.8
(2x−3)23=4
Exercise 10.7.9
(x+12)−23=4
Exercise 10.7.10
(x−1)−52=32
Exercise 10.7.11
(3x−2)45=16
Exercise 10.7.12
(4x+2)35=−8
Exercise 10.7.13
x3=−8
Exercise 10.7.14
4x3−2=106
Exercise 10.7.15
(x−4)2=49
Exercise 10.7.16
(5x+1)4=16
Exercise 10.7.17
(2x+1)2+3=21
Exercise 10.7.18
(x−1)32=8
Exercise 10.7.19
(2x+3)43=16
Exercise 10.7.20
(x+3)−13=4
Exercise 10.7.21
(x−1)−53=32
Exercise 10.7.22
(x+3)32=−8
Exercise 10.7.23
(2x+3)32=27
Exercise 10.7.24
(3−2x)43=−81