Question

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Part 1
Find all the real fourth roots of 1616.

Answer 1

To find all the real fourth roots of a number, we can use the formula for the \( n \)-th roots of a number. The fourth roots of a number \( a \) can be expressed as:

\[
x = a^{1/4}
\]

where the roots can be calculated using both positive and negative values.

We start by calculating the fourth root of 1616:

1. Calculate \( 1616^{1/4} \):

First, we will find the approximate value of \( 1616^{1/4} \) using a calculator.

\[
1616^{1/4} \approx 5.44
\]

2. Now, considering the nature of roots, the fourth roots of a positive number like 1616 will include both positive and negative values. Fourth roots can be expressed in general as:

\[
x = \sqrt[4]{1616} \quad \text{and} \quad -\sqrt[4]{1616}
\]

Thus, we have:

\[
x_1 = 5.44 \quad \text{and} \quad x_2 = -5.44
\]

In conclusion, the real fourth roots of 1616 are approximately:

\[
\boxed{5.44} \quad \text{and} \quad \boxed{-5.44}
\]