Question
Find all the real fourth roots of 1616.
Question content area bottom
Part 1
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The real fourth root(s) of 1616 is/are enter your response here.
(Simplify your answer. Use a comma to separate answers as needed.)
B.
There are no real fourth roots of 1616.
Question content area bottom
Part 1
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The real fourth root(s) of 1616 is/are enter your response here.
(Simplify your answer. Use a comma to separate answers as needed.)
B.
There are no real fourth roots of 1616.
Answers
GPT-4o mini
To find the real fourth roots of \( 1616 \), we can express this mathematically as:
\[
x^4 = 1616
\]
To find the fourth roots, we take the fourth root of both sides:
\[
x = \sqrt[4]{1616}
\]
Calculating \( \sqrt[4]{1616} \):
First, we can simplify \( 1616 \) by factorizing it:
1. \( 1616 \) is even, so we divide by \( 2 \):
\[
1616 \div 2 = 808
\]
2. \( 808 \) is also even, so divide by \( 2 \):
\[
808 \div 2 = 404
\]
3. \( 404 \) is even, so divide by \( 2 \):
\[
404 \div 2 = 202
\]
4. \( 202 \) is even, so divide by \( 2 \):
\[
202 \div 2 = 101
\]
5. \( 101 \) is a prime number.
Thus, we can write the factorization of \( 1616 \) as:
\[
1616 = 2^4 \times 101
\]
Now we can compute the fourth root:
\[
\sqrt[4]{1616} = \sqrt[4]{2^4 \times 101} = \sqrt[4]{2^4} \times \sqrt[4]{101} = 2 \times \sqrt[4]{101}
\]
Since \( \sqrt[4]{101} \) is a positive number, the real fourth root of \( 1616 \) is:
\[
2 \sqrt[4]{101}
\]
Identifying the real fourth roots, we also consider that the equation \( x^4 = 1616 \) has four roots in total, including negative ones. The four roots are:
\[
2 \sqrt[4]{101}, -2 \sqrt[4]{101}, 2i \sqrt[4]{101}, -2i \sqrt[4]{101}
\]
But however, we are only interested in the real roots, which are:
\[
2 \sqrt[4]{101} \quad \text{and} \quad -2 \sqrt[4]{101}
\]
Thus, the final answers and the choice are:
A. The real fourth root(s) of \( 1616 \) is/are: \( 2 \sqrt[4]{101}, -2 \sqrt[4]{101} \) (where \( \sqrt[4]{101} \) can be approximated numerically if needed).
\[
x^4 = 1616
\]
To find the fourth roots, we take the fourth root of both sides:
\[
x = \sqrt[4]{1616}
\]
Calculating \( \sqrt[4]{1616} \):
First, we can simplify \( 1616 \) by factorizing it:
1. \( 1616 \) is even, so we divide by \( 2 \):
\[
1616 \div 2 = 808
\]
2. \( 808 \) is also even, so divide by \( 2 \):
\[
808 \div 2 = 404
\]
3. \( 404 \) is even, so divide by \( 2 \):
\[
404 \div 2 = 202
\]
4. \( 202 \) is even, so divide by \( 2 \):
\[
202 \div 2 = 101
\]
5. \( 101 \) is a prime number.
Thus, we can write the factorization of \( 1616 \) as:
\[
1616 = 2^4 \times 101
\]
Now we can compute the fourth root:
\[
\sqrt[4]{1616} = \sqrt[4]{2^4 \times 101} = \sqrt[4]{2^4} \times \sqrt[4]{101} = 2 \times \sqrt[4]{101}
\]
Since \( \sqrt[4]{101} \) is a positive number, the real fourth root of \( 1616 \) is:
\[
2 \sqrt[4]{101}
\]
Identifying the real fourth roots, we also consider that the equation \( x^4 = 1616 \) has four roots in total, including negative ones. The four roots are:
\[
2 \sqrt[4]{101}, -2 \sqrt[4]{101}, 2i \sqrt[4]{101}, -2i \sqrt[4]{101}
\]
But however, we are only interested in the real roots, which are:
\[
2 \sqrt[4]{101} \quad \text{and} \quad -2 \sqrt[4]{101}
\]
Thus, the final answers and the choice are:
A. The real fourth root(s) of \( 1616 \) is/are: \( 2 \sqrt[4]{101}, -2 \sqrt[4]{101} \) (where \( \sqrt[4]{101} \) can be approximated numerically if needed).