Question

find all the real fourth roots of 81.

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the real fourth root(s) of 81 is/are .
(simplify your answer. use a comma to separate answers as needed.)

b. there are no real fourth roots of 81.

Explanation:

Step1: Recall the definition of nth root

A real fourth root of a number \( x \) is a real number \( a \) such that \( a^4=x \). So we need to find all real numbers \( a \) where \( a^4 = 81 \).

Step2: Solve the equation \( a^4=81 \)

We can rewrite the equation as \( a^4 - 81=0 \). Notice that \( a^4 - 81=(a^2)^2-9^2 \), and using the difference of squares formula \( x^2 - y^2=(x - y)(x + y) \), we get \( (a^2 - 9)(a^2+9)=0 \).

Now, factor \( a^2 - 9 \) further (again using difference of squares): \( (a - 3)(a + 3)(a^2+9)=0 \).

We set each factor equal to zero:

• For \( a - 3 = 0 \), we get \( a = 3 \).
• For \( a + 3 = 0 \), we get \( a=- 3 \).
• For \( a^2+9 = 0 \), we have \( a^2=-9 \), and since the square of a real number cannot be negative, there are no real solutions from this factor.

So the real solutions to \( a^4 = 81 \) are \( a = 3 \) and \( a=-3 \).

Answer:

A. The real fourth root(s) of 81 is/are \( 3, - 3 \).