Question

find all the real fourth roots of 2401.

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

a. the real fourth root(s) of 2401 is/are .
(simplify your answer. use a comma to separate answers as needed.)

b. there are no real fourth roots of 2401.

Explanation:

Step1: Recall the definition of fourth root

A real fourth root of a number \( x \) is a real number \( a \) such that \( a^4=x \). So we need to find real numbers \( a \) where \( a^4 = 2401 \).

Step2: Analyze the sign and find possible roots

We know that for real numbers, if we raise a positive number to the fourth power, we get a positive number, and if we raise a negative number to the fourth power, we also get a positive number (since the exponent is even). So we can check positive and negative numbers.

First, let's find a positive number \( a \) such that \( a^4=2401 \). Let's start by checking small integers. We know that \( 7^2 = 49 \), then \( 7^4=(7^2)^2=49^2 = 2401 \). Also, \( (- 7)^4=[(-7)^2]^2 = 49^2=2401 \).

Answer:

A. The real fourth root(s) of 2401 is/are \( 7, - 7 \).