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Chapter 7: Problem 70

Simplify. \(\sqrt[6]{1458}\)

Short Answer

Expert verified
Simplified result is 3.

Step by step solution

01

- Prime Factorization

Find the prime factorization of 1458. Start by dividing the number by the smallest prime number until you reach 1.
02

- Group the Factors

Group the prime factors such that each group contains six identical factors, because we are dealing with the 6th root.
03

- Take the 6th Root

Take one factor out of each group. This is because the 6th root of a product of six identical factors is just that factor.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Factorization
Understanding the prime factorization of a number is essential for many mathematical concepts, including simplifying roots.
Prime factorization means breaking down a number into the product of its prime numbers.
A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.
For example, to find the prime factorization of 1458, you start by dividing by the smallest prime number, 2:
1458 梅 2 = 729.
Now, divide by the next smallest prime number, which is still 3:
729 梅 3 = 243.
Keep dividing by 3 until you can't anymore. So:
243 梅 3 = 81,
81 梅 3 = 27,
27 梅 3 = 9,
9 梅 3 = 3,
3 梅 3 = 1.
So, the prime factorization of 1458 is \(2 \times 3^6\).
Sixth Root
The sixth root simply means that we are looking for a number which, when multiplied by itself six times, gives us the original number.
In this case, we want to find the sixth root of 1458, represented mathematically as \(\root {6} \) or \(\root {6} {1458}\).
Since we have already performed prime factorization and know that 1458 = 2 脳 3^6, we can now use this information to find the sixth root
The sixth root of 3^6 is just 3 because: \(3 ^ 6 = 729\).
Simplification Steps
Simplifying higher-order roots involves several logical steps.
Here are the streamlined simplification steps for our problem:

1. **Prime Factorization:** As seen, we first break down 1458 into its prime factors of 2 and 3. So, 1458 = 2 脳 3^6.

2. **Group the Factors:** Since we are dealing with the 6th root, we group the prime factors into groups of six. In this case, we get exactly one group of \(3^6\).

3. **Take the 6th Root:** Taking the 6th root of \(3^6\) leaves us with 3. Therefore, the sixth root of 1458 is simply 3.
Always remember to deal with each factor group separately and note that the number outside the root is the power we reduce by.

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