Exercise 33 Page 266

To solve equations with a variable expression raised to a rational exponent, we raise each side of the equation to the reciprocal of the rational exponent. x^(m n)=k ⇔ (x^(m n))^() n m=k^() n m Remember, if m is even, then (x^(m n))^() n m=|x|. In this case m= 1, so we do not need to worry about this. Let's raise each side of the equation to the power of 2. We obtained a quadratic equation. There are many ways to solve a quadratic equation. We will use the Quadratic Formula to solve the equation. Let's determine a, b, and c. x^2+2x-35=0 ⇔ 1x^2+ 2x+( - 35)=0 We see above that a= 1, b= 2, and c= - 35. Let's substitute these values into the Quadratic Formula to solve the equation.

x=- b±sqrt(b^2-4ac)/2a

SubstituteValues

Substitute values

x=- 2±sqrt(2^2-4( 1)( - 35))/2( 1)

▼

Simplify right-hand side

CalcPow

Calculate power

x=- 2±sqrt(4-4(1)(- 35))/2(1)

Multiply

Multiply

x=- 2±sqrt(4+140)/2

AddTerms

Add terms

x=- 2±sqrt(144)/2

CalcRoot

Calculate root

x=- 2± 12/2

Now we can calculate the first root using the positive sign and the second root using the negative sign.

x=- 2± 12/2
x=- 2-12/2	x=- 2+12/2
x=- 7	x=5

Checking the Solutions

Next, we will check the solutions by substituting - 7 and 5 for x into the original equation. If the substitution produces a true statement, we know that our answer is correct. If it does not, then it is an extraneous solution. Let's first check the - 7.

2(x+11)^(12)=x+3

Substitute

x= - 7

2( - 7+11)^(12)? = - 7+3

▼

Simplify

AddTerms

Add terms

2(4)^(12)? =- 4

WritePow

Write as a power

2(2^2)^(12)? =-4

PowPow

(a^m)^n=a^(m* n)

2(2)? =-4

Multiply

Multiply

4≠ - 4 *

Because our substitution produced a false statement, we know that our answer, x=- 7, is not correct. Let's check now 5.

2(x+11)^(12)=x+3

Substitute

x= 5

2( 5+11)^(12)? = 5+3

▼

Simplify

AddTerms

Add terms

2(16)^(12)? =8

WritePow

Write as a power

2(4^2)^(12)? =8

PowPow

(a^m)^n=a^(m* n)

2(4)? =8

Multiply

Multiply

8=8 ✓

Because our substitution produced a true statement, we know that our answer, x=5, is correct.