Problem 92 - Algebra 1

Question

SIMPLIFYING EXPRESSIONS Simplify. Write your answer as a power or as an expression containing powers. $$\left(3 b^{4}\right)^{2}$$

Step-by-Step Solution

Answer

The simplified expression is \(9b^{8}\).

Step 1: Identify the Inner Operation

Start by identifying the inner operation in the given expression. Here, \(3b^{4}\) is considered as a single entity because it's within the parentheses.

Step 2: Apply the Power of a Power Rule

The power of a power rule states that \((a^{m})^{n}=a^{m \times n}\). Use this rule to rewrite the expression. In this case, \(3b^{4}\) is raised to the power 2.

Step 3: Distribute the Power

Apply the power to both components (3 and \(b^{4}\)) inside the brackets. This is because the distributive law of exponentiation over multiplication states that \((ab)^{n}=a^{n}b^{n}\).Apply this rule so to separately raise both 3 and \(b^{4}\) to the power of 2.

Step 4: Simplify the Result

Simplify the thus far calculated expression. In this case it means calculating \(3^{2}\) and multiplying the exponents in \(b^{4 \times 2}\).

Key Concepts

Power of a Power Rule

The power of a power rule is a fundamental concept in exponentiation. It allows you to simplify expressions where an exponent is raised to another exponent. The rule is stated as \((a^m)^n = a^{m \times n}\). Therefore, you're essentially multiplying the exponents.

Let's look at an example to clarify:

• Consider \((b^4)^2\), where \(b\) is raised first to the 4th power and then the result is squared.
• According to the power of a power rule, we multiply the exponents: \(4 \times 2\).
• This simplifies to \(b^8\).

Remember, only the exponents are multiplied, the base \(b\) remains unchanged.

This rule is particularly useful when dealing with nested exponents and helps simplify complex expressions quickly. It reduces them into their simplest forms while maintaining mathematical accuracy.

Distributive Law of Exponentiation

The distributive law of exponentiation, also known as the Product of Powers rule, involves distributing an outer exponent to each component within a product inside a set of parentheses. This can be expressed as \((ab)^n = a^n b^n\).

In simpler terms, if a product is raised to a power, both factors receive the exponent.

• For example, consider \((3b^4)^2\).
• Using the distributive law, we separately apply the exponent of 2 to both \(3\) and \(b^4\).
• This means calculating \(3^2\) and \((b^4)^2\), resulting in \(9\) and \(b^8\) respectively.

It’s important to note that you distribute the exponent to each factor inside the brackets. This allows you to simplify expressions accurately and efficiently. By doing so, you maintain the structural integrity of the expression while breaking it down into manageable parts.

Simplifying Expressions

Simplifying expressions involves using mathematical rules and principles to reduce complex expressions to their simplest forms. After applying the power of a power rule and the distributive law of exponentiation, the next step is to perform calculations and ensure everything is in its simplest state.

With the expression \((3b^4)^2\), first, apply both rules:

• Calculate \(3^2\), resulting in \(9\).
• Calculate \((b^4)^2\) by using the power of a power rule, which gives \(b^8\).

Finally, write the expression as \(9b^8\).

The key focus in simplifying is breaking down the expression systematically, ensuring each rule and calculation is correctly applied. This not only provides a clear solution but also strengthens your understanding of the underlying math principles.