Problem 109 - Elementary and Intermediate Algebra

Question

Simplify each radical expression, if possible. Assume all variables are unrestricted. $$ \sqrt[3]{-125 m^{6}} $$

Step-by-Step Solution

Answer

The simplified expression is \\-5m^2\\.

Step 1: Identify the Cube Root Components

First, recognize that the expression inside the cube root, \(-125 m^6\), can be separated into two parts: a constant \(-125\) and a variable \(m^6\). The cube root of a product can be expressed as the product of cube roots: \(\sqrt[3]{-125}\times\sqrt[3]{m^6}\).

Step 2: Cube Root of the Constant Term

Determine the cube root of \(-125\). Since \(-125 = (-5)^3\), we can simplify \((\sqrt[3]{-125})\) to \-5\.

Step 3: Cube Root of the Variable Term

For the variable \(m^6\), find the cube root by dividing the exponent by 3: \((\sqrt[3]{m^6} = m^{6/3} = m^2)\).

Step 4: Simplify the Expression

Combine the results from Step 2 and Step 3 to simplify the original expression. Thus, \(\sqrt[3]{-125 m^6}\) simplifies to \-5m^2\.

Key Concepts

Cube Root

The cube root of a number is an operation that reverses the action of cubing. In other words, if you have a number \(a\) such that \(a^3 = b\), then \(a\) is the cube root of \(b\). The cube root is denoted using the radical symbol with a small 3 in the index, like this: \(\sqrt[3]{b}\). This is different from a square root, which only involves pairs, as cube roots work on groups of three.

A cube root can be split into separate parts if necessary. In the original example, \(\sqrt[3]{-125 m^6}\), the inside of the root was split into \(-125\) and \(m^6\). Each part can be evaluated separately using the cube root.

• The cube root of a negative number results in a negative number. Thus, \(\sqrt[3]{-125} = -5\) because \((-5)\times(-5)\times(-5) = -125\).
• The cube root of \(m^6\) involves working with the exponents, which will be discussed in the next section.

Exponent Rules

Understanding how exponents work is crucial when dealing with cube roots and simplifying expressions. Exponents tell us how many times a number is multiplied by itself. The basic rule to remember is when finding the root of a power, you divide the exponent by the root's index.

For example, in \(\sqrt[3]{m^6}\):

• The number "6" is the exponent of \(m\).
• To get the cube root, divide this exponent by 3 (the cube root's index): \(\frac{6}{3} = 2\).
• This results in \(m^2\).

These rules apply consistently across various root and exponent situations. Additionally, negative bases with odd exponents remain negative. When multiplying powers with the same base, add the exponents. For division, subtract them.

Simplification of Expressions

Simplifying expressions is the process of rewriting them in their simplest form. This involves both the numerical and the variable components. In the given example, \(\sqrt[3]{-125 m^6}\), the simplification starts with splitting the expression into parts: the constant \(-125\) and the variable \(m^6\). Each is simplified separately:

• The cube root of \(-125\) becomes \(-5\) based on the cube root rule.
• The cube root of \(m^6\) becomes \(m^2\) using the exponent rules.

Once you find the simplified versions of each part, combine them. This gives you \(-5m^2\) as the simplified expression. This process helps to handle more complex problems by breaking them down into easier steps.

Remember, simplification is about making expressions as straightforward as possible without changing their value. So, taking each part step by step ensures clarity and correctness.