Problem 51 - Algebra A Combined Function

Question

Simplify. Assume that the variables represent any real number. $$ \sqrt[3]{(-8)^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is -8.

Step 1: Identify the Expression

The expression we need to simplify is \( \sqrt[3]{(-8)^3} \). This is a cube root applied to the cube of -8.

Step 2: Use the Property of Cube Roots

Recall that the cube root of a number cubed is simply the original number. Mathematically, this is expressed as \( \sqrt[3]{a^3} = a \), assuming \( a \) is any real number.

Step 3: Simplify Using the Property

Apply the property \( \sqrt[3]{(-8)^3} = -8 \). Here, \( a \) is equal to -8, so the cube root of \((-8)^3\) is -8.

Key Concepts

Simplification

Simplification in mathematics refers to the process of reducing expressions into their simplest or most understandable form. The aim is to make complex expressions easier to work with or interpret. This is often done by using rules, properties, and strategies that streamline operations:

• Breaking down expressions using known mathematical properties, like distributive or associative laws.
• Identifying and cancelling common factors or like terms.
• Rewriting expressions using simpler numbers or terms.

In the exercise provided, we are asked to simplify \( \sqrt[3]{(-8)^{3}} \). Here, the goal is to recognize that calculating the cube of -8 and then taking the cube root are inverse operations. By applying the property \( \sqrt[3]{a^3} = a \), the expression simplifies directly to -8, skipping complex calculations.

Real Numbers

Real numbers are a fundamental concept in mathematics, encompassing all the numbers on the number line. This includes:

• Natural numbers (1, 2, 3, ...).
• Whole numbers (0, 1, 2, 3, ...).
• Integers (-3, -2, -1, 0, 1, 2, 3, ...).
• Rational numbers (fractions like \( \frac{1}{2} \), \( \frac{3}{4} \) and so forth).
• Irrational numbers (numbers that cannot be expressed as fractions, such as \( \sqrt{2} \), \( \pi \)).

Real numbers can be positive, negative, or zero, and they can be used to measure continuous quantities. In the context of the exercise, any value, including -8, is considered a real number. The rule \( \sqrt[3]{a^3} = a \) is valid for all real numbers, showcasing a broad applicability that makes this simplification method reliable.

Properties of Exponents

Exponents are a critical component in mathematics, allowing us to express repeated multiplication concisely. The properties of exponents simplify complex expressions by following specific rules, such as:

• Power of a power: \( (a^m)^n = a^{m \times n} \)
• Product of powers: \( a^m \times a^n = a^{m+n} \)
• Quotient of powers: \( \frac{a^m}{a^n} = a^{m-n} \) if \( a eq 0 \)

In the given exercise, we particularly use the concept of a power involving a cube, \( a^3 \), which reverses when taking a cube root. This property \( \sqrt[3]{a^3} = a \) shows how exponents' properties help in simplifying expressions. They allow us to focus on the base, which, in our exercise's case, quickly translates \( \sqrt[3]{(-8)^3} \) to \( -8 \), emphasizing the power-exponent inverse relationship.