Problem 467 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\sqrt[3]{625}\) (b) \(\sqrt[6]{128}\)

Step-by-Step Solution

Answer

(a) \(5^{4/3}\); (b) \(2^{7/6}\)

Step 1: Understanding Cube Root

To simplify \(\sqrt[3]{625}\), we need to find a number which, when cubed, gives 625.

Step 2: Factorize the Number

625 can be factorized as follows: \(625 = 5\times 5\times 5\times 5 = 5^4\).

Step 3: Simplify Cube Root

The cube root of \(5^3\) is clearly 5, since \(5^3 = 125\). But we need to cube root 625. Since \(625 = 5^4\), we identify that \(\sqrt[3]{5^4} = 5^{4/3}\).

Step 4: Simplified Answer for (a)

Therefore, \(\sqrt[3]{625} = 5^{4/3} \). This is the simplest form we will have without a single integer value.

Step 5: Understanding Sixth Root

To simplify \(\sqrt[6]{128}\), we need to find a number which, when raised to the power of 6, gives 128.

Step 6: Factorize the Number

128 can be factorized as follows: \(128 = 2\times 2\times 2\times 2\times 2\times 2\times 2 = 2^7\).

Step 7: Simplify Sixth Root

The sixth root of \(2^6\) is clearly 2, since \(2^6 = 64\). But we need to sixth root 128. Since \(128 = 2^7\), we identify that \(\sqrt[6]{2^7} = 2^{7/6}\).

Step 8: Simplified Answer for (b)

Therefore, \(\sqrt[6]{128} = 2^{7/6} \). This is the simplest form we will have without a single integer value.

Key Concepts

Cube Roots

Cube roots help us find a number which, when raised to the power of three (cubed), equals the given number. Think of it as the opposite of cubing a number. When simplifying cube roots, the first step is to factorize the given number.
For example, \( \sqrt[3]{625} \) can be factorized as follows:\