Problem 45 - Elementary and Intermediate Algebra

Question

Square or cube each quantity and simplify the result. $$ (\sqrt[3]{12})^{3} $$

Step-by-Step Solution

Answer

The result of \((\sqrt[3]{12})^{3}\) is 12.

Step 1: Understand the Problem

We have an expression \((\sqrt[3]{12})^{3}\). Our goal is to simplify this expression.

Step 2: Apply the Cube Property

We know from exponentiation rules that \((\sqrt[3]{x})^{3} = x\) for any number \(x\). In this case, \(x = 12\). Therefore, applying this property, we get: \((\sqrt[3]{12})^{3} = 12\).

Step 3: Simplify the Expression

After applying the cube property, the expression simplifies directly to 12. There are no further steps since the expression is already in its simplest form.

Key Concepts

Cube Roots

A cube root is the number that, when multiplied by itself three times, gives the original number. It's the inverse operation of raising a number to the power of three. For example, the cube root of 8 is 2 because when we calculate \( 2 \times 2 \times 2 \), we get 8. In mathematical notation, the cube root of a number \( x \) is written as \( \sqrt[3]{x} \).

Finding cube roots can seem daunting at first, but once you understand how it works, it becomes straightforward:

• Think of it like "un-cubing" a number.
• Cubing a number and then taking the cube root will bring you back to your original number.
• In our exercise, taking the cube root of 12 and then raising it to the power of 3 brings us back to 12.

Having a strong grasp of cube roots aids in solving more complicated algebraic equations. Knowing that applying the cube root and then its inverse operation (cubing) results in the original number simplifies calculations significantly.

Exponentiation Rules

Understanding exponentiation rules is crucial for simplifying expressions involving powers and roots. The rules help determine how to manipulate expressions with exponents effectively. Here's what you need to know:

When dealing with cube roots and exponents, remember that raising a power to another power involves multiplication of the exponents. This is shown in the rule \((x^a)^b = x^{a \cdot b}\). When the base is a radical like \(\sqrt[3]{x}\), things are no different:

• \((\sqrt[3]{x})^3 = (x^{1/3})^3 = x^{1/3 \cdot 3} = x^1 = x\).
• The exponent \(1/3\) is the cube root, and multiplying by 3 effectively cancels it out.

Exponentiation rules simplify tasks such as multiplication, division, and conversion between radicals and exponents. Recognizing how powers and roots operate in algebra is important for simplifying the expressions like the one in our exercise.

Algebraic Simplification

Algebraic simplification means reducing an expression to its simplest form. This involves using operations like addition, subtraction, multiplication, and division without changing the value. Simplicity makes expressions easier to understand and solve. In our example, we simplified \((\sqrt[3]{12})^3\) to 12 using properties of exponents and cube roots.

Here are some key pointers on simplification:

• Identify parts of the expression that can be simplified using basic rules (like exponentiation rules).
• Always aim for the cleanest most straightforward expression.
• Check if there are common factors, powers, or similar components that can be manipulated.

Simplifying an expression sheds light on its underlying structure, often revealing its simplest form with minimal computation. Mastering algebraic simplification is instrumental in solving complex equations and assists in building a solid algebraic foundation.