Problem 55 - Elementary Algebra

Question

Simplify \(\left(a^{3} b^{6}\right)^{4}\)

Step-by-Step Solution

Answer

Question: Simplify the expression \(\left(a^{3} b^{6}\right)^{4}\) using the properties of exponents. Answer: \(a^{12} b^{24}\)

Step 1: Apply the power of a product property

Using the power of a product property, distribute the exponent 4 to each term inside the parenthesis: \((a^3 b^6)^4 = a^3{}^4 b^6{}^4\).

Step 2: Apply the power of a power property

Apply the power of a power property by multiplying the exponents: \(a^3{}^4 = a^{3 \cdot 4}\), and \(b^6{}^4 = b^{6 \cdot 4}\).

Step 3: Multiply the exponents

Multiply the exponents to get the simplified form: \(a^{3 \cdot 4} = a^{12}\), and \(b^{6 \cdot 4} = b^{24}\).

Step 4: Write the final answer

Combine the simplified terms to get the final answer: \(\left(a^{3} b^{6}\right)^{4} = a^{12} b^{24}\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, called the base, to a power, known as the exponent. When you see a number written as \(a^n\), it means that the base \(a\) is multiplied by itself \(n\) times.
This operation is fundamental in algebra and is used for expressing repeated multiplication in a compact form.
For instance, \(a^3\) means \(a \times a \times a\).

• The base is the number being multiplied.
• The exponent tells us how many times to multiply the base by itself.

In expressions like \((a^{3} b^{6})^{4}\), exponentiation is applied twice: first to each base internally and then to the entire product as a whole. This use of exponentiation simplifies complex expressions, making calculations and interpretations easier.

Properties of Exponents

Understanding and applying the properties of exponents can greatly simplify complex algebraic expressions. One of the most used properties is the power of a product property.
This property states that when raising a product to a power, each factor in the product is raised to the power individually. In other words, \((xy)^n = x^n y^n\).
Another key property is the power of a power property. This determines how to handle nested powers, showing that to simplify \((a^m)^n\), you multiply the exponents: \((a^m)^n = a^{m \cdot n}\).

• Power of a Product: \((ab)^n = a^n b^n\)
• Power of a Power: \((a^m)^n = a^{m \cdot n}\)

When applied to our expression \((a^{3} b^{6})^{4}\), these properties allow us to break down the expression systematically, leading to a simplified result.

Simplification

The process of simplification in mathematics involves making expressions or computations easier to manage by condensing them into simpler or more compact forms. With algebraic expressions, simplification requires the application of algebraic rules, such as the properties of exponents.
In the context of the exercise \((a^{3} b^{6})^{4}\), simplification involves applying the power of a product and power of a power properties consecutively.

• Distributing exponents across the terms: \((a^3 b^6)^4 = a^{3 \cdot 4} b^{6 \cdot 4}\)
• Performing the multiplication: \(a^{12} b^{24}\)

By methodically applying these steps, the expression is simplified, resulting in a clearer and more understandable form for further mathematical use or analysis.