Raise each side of the equation to the reciprocal of the rational exponent.
x=1
Practice makes perfect
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.
We will first isolate the term with the rational exponent, and then raise each side of the equation to the power of 2.
We obtained a quadratic equation. Notice that there are many ways to solve a quadratic equation. We will use the Quadratic Formula to solve this equation. Let's determine a, b, and c.
4x^2+x-5=0 ⇔ 4x^2+ 1x+( - 5)=0
We see above that a= 4, b= 1, and c= - 5. Let's substitute these values into the Quadratic Formula to solve the equation.
Now we can calculate the first root using the positive sign and the second root using the negative sign.
x=- 1± 9/8
x=- 1-9/8
x=- 1+9/8
x=-5/4
x=1
Checking the Solutions
Next, we will check the solutions by substituting - 54 and 1 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 - 54.
