Evaluate ³√−512 = and ⁴√6561 =
Understand the Problem
The question is requesting the evaluation of two expressions involving roots: the cube root of -512 and the fourth root of 6561.
Answer
The answers are $-8$ and $9$.
Answer for screen readers
The cube root of -512 is $-8$ and the fourth root of 6561 is $9$.
Steps to Solve
- Evaluating the Cube Root of -512
To find the cube root of -512, we can express it as: $$ \sqrt[3]{-512} $$
We know that the cube root of a negative number results in a negative root. Thus, we can find the cube root of 512 first: $$ \sqrt[3]{512} = 8 $$
Therefore, $$ \sqrt[3]{-512} = -8 $$
- Evaluating the Fourth Root of 6561
Next, we evaluate the fourth root of 6561: $$ \sqrt[4]{6561} $$
We start by finding the prime factorization of 6561: $$ 6561 = 3^8 $$
So, we can rewrite the fourth root as: $$ \sqrt[4]{3^8} = 3^{8/4} = 3^2 $$
Calculating this gives: $$ 3^2 = 9 $$
The cube root of -512 is $-8$ and the fourth root of 6561 is $9$.
More Information
The cube root and fourth root can be used in various applications, such as solving equations and analyzing geometric properties. Recognizing the relationship between powers and roots is essential in algebra.
Tips
- Confusing even roots (like the fourth root) with odd roots (like the cube root). Remember that negative numbers don't have real even roots.
- Forgetting to consider the fact that $-512$ has a negative cube root.
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