Problem 445 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\sqrt[3]{125}\) (b) \(\sqrt[4]{1296}\) \(\sqrt[5]{1024}\)

Step-by-Step Solution

Answer

The simplified forms are 5, 6, and 4 respectively.

Step 1: Simplify \(\textbackslash sqrt[3]{125}\)

First, recognize that 125 is a perfect cube. Identify that 125 can be written as \(5^3\). Therefore, \(\textbackslash sqrt[3]{125} = 5\).

Step 2: Simplify \(\textbackslash sqrt[4]{1296}\)

Next, recognize that 1296 is a perfect fourth power. Identify that 1296 can be written as \(6^4\). Therefore, \(\textbackslash sqrt[4]{1296} = 6\).

Step 3: Simplify \(\textbackslash sqrt[5]{1024}\)

Lastly, recognize that 1024 is a perfect fifth power. Identify that 1024 can be written as \(4^5\). Therefore, \(\textbackslash sqrt[5]{1024} = 4\).

Key Concepts

Perfect Cube

A perfect cube is a number that can be written as the cube of an integer. In other words, it's a number that is the result of multiplying an integer by itself twice. For instance, the number 125 is a perfect cube because it can be expressed as \(5^3\), which is equivalent to \(5 \times 5 \times 5 = 125\).

To simplify a cube root of a number, you need to check if the number itself is a perfect cube. If it is, identify which integer, when raised to the power of three, gives you the original number.

For example:

• \(\sqrt[3]{27} = 3\) because \(3^3 = 27\).


• \(\sqrt[3]{64} = 4\) because \(4^3 = 64\).
.

This process involves recognizing patterns and being comfortable with basic cube values.

Perfect Fourth Power

A perfect fourth power is a number that can be expressed as a number raised to the power of four. Essentially, it's the outcome of multiplying an integer by itself three more times. For instance, 1296 is a perfect fourth power because it can be written as \(6^4\).

This means that \(6 \times 6 \times 6 \times 6 = 1296\). To simplify the fourth root of a number, figure out whether the number is a perfect fourth power. If it is, determine the integer that, when raised to the fourth power, equals the original number.

For example:

• \(\sqrt[4]{256} = 4\) because \(4^4 = 256\)


• \(\sqrt[4]{16} = 2\) because \(2^4 = 16\)

Identifying these patterns helps in quickly simplifying roots that appear in problems.

Perfect Fifth Power

A perfect fifth power is a number that can be expressed as an integer raised to the power of five. In simpler terms, it's a number that results from multiplying an integer by itself four additional times. For example, 1024 is a perfect fifth power because it can be written as \(4^5\).

This means that \(4 \times 4 \times 4 \times 4 \times 4 = 1024\). To simplify the fifth root of a number, check if it is a perfect fifth power. If the number is, you must identify the integer that, when raised to the power of five, gives the original number.

For example:

• \(\sqrt[5]{32} = 2\) because \(2^5 = 32\)


• \(\sqrt[5]{243} = 3\) because \(3^5 = 243\)

Understanding these relationships can facilitate the process of simplifying higher-order roots in mathematics.