Problem 39 - Algebra 2

Question

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[3]{0.125} $$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt[3]{0.125}\) is \(0.5\).

Step 1: Recognize 0.125 as a fraction

\(0.125 = \frac{125}{1000} = \frac{1}{8}\)

Step 2: Compute the cube root

\(\sqrt[3]{0.125} = \sqrt[3]{\frac{1}{8}} = \frac{\sqrt[3]{1}}{\sqrt[3]{8}} = \frac{1}{2} = 0.5\)

Key Concepts

Cube Roots

Cube roots are an important concept in algebra that deals with finding a number which, when multiplied by itself three times, equals the given number. So, if you have a number and you want to find its cube root, you are looking for a value that, when used in a multiplication three times, gives you that number.
For example, to find the cube root of 8, you need a number which satisfies the equation:

• \( x^3 = 8 \)

By testing, you find that 2 works:

• \( 2 \times 2 \times 2 = 8 \), therefore, the cube root of 8 is 2.

In a similar way, for our exercise number 0.125, we need to find a number who satisfies:

• \( x^3 = 0.125 \)

Here, by multiplication, we find:

• \( 0.5 \times 0.5 \times 0.5 = 0.125 \).

Hence, the cube root of 0.125 is 0.5. Cube roots are denoted by the radical symbol and a small 3, like this:

• \( \sqrt[3]{x} \).

Radical Expressions

Radical expressions involve roots, symbolized as radicals (\(\sqrt{}\)). While this symbol often stands for the square root, in the context of algebra, it can represent any root, including cube roots.
Essentially, when you're asked to simplify a radical expression, your task is to express it in the simplest form possible. This involves finding the smallest possible base number for a given exponent within the root.

• For instance, simplifying \(\sqrt[3]{0.125}\) means we transform the expression to its simplest form: 0.5.

Simplifying radical expressions has various applications in algebra, making complex problems easier to solve. It helps in reducing computation time and helps in recognizing equivalences, leading to more straightforward solutions.

Absolute Value in Algebra

In algebra, the absolute value of a number refers to its distance from zero on the number line, without considering direction. It’s always a positive number or zero.
We apply absolute value in algebra when we simplify expressions, especially when dealing with variables that can have both negative and positive values. Absolute values ensure that the expression only has positive outcomes, regardless of the sign of the input.

• For example, the absolute value of -3 is 3, because it is three units away from 0 on the number line.
• Similarly, \(|-x| = x\) if x is positive.

When simplifying cube roots, absolute values are less frequently needed, as cube roots handle negative numbers by returning negative roots of negative numbers, unlike square roots.