Problem 88 - Algebra A Combined Function

Question

Simplify each expression. $$ (2 a b)^{4} $$

Step-by-Step Solution

Answer

The expression simplifies to \(16a^4b^4\).

Step 1: Understanding the Expression

We need to simplify the expression \((2ab)^4\). This expression means that the entire term inside the parentheses, \(2ab\), is raised to the power of 4.

Step 2: Apply the Power to Each Component

According to the exponentiation power rule, \((xy)^n = x^n y^n\). Therefore, \((2ab)^4 = 2^4 a^4 b^4\). This means we need to calculate \(2^4\) and apply the power of 4 to both \(a\) and \(b\).

Step 3: Calculate the Numerical Power

Let's calculate \(2^4\). This means \(2 imes 2 imes 2 imes 2 = 16\). Thus, \(2^4 = 16\).

Step 4: Write the Simplified Expression

Substitute \(16\) back into the expression. Thus, the simplified version of \((2 ab)^4\) is \(16a^4b^4\).

Key Concepts

Exponentiation

Exponentiation is a fundamental concept in algebra that involves raising a number or expression to a certain power. The base tells us what number or expression we're repeatedly multiplying, while the exponent tells us how many times to multiply it. In the expression \[ (2ab)^4 \]\(2ab\) is the base and 4 is the exponent. This tells us to multiply \(2ab\) by itself 4 times.

• When you see something raised to a power, like \((xy)^n\), it means that each factor in the base must be multiplied by itself, \(n\) times.

Understanding this concept is crucial, as it applies not only to numbers, but to algebraic expressions as well. This building block helps pave the way for more complex mathematical operations.

Simplification

Simplification in algebra involves breaking down complex expressions into simpler or more manageable forms, often making equations easier to solve or manipulate. When simplifying,

• We aim to combine like terms or apply rules to clear complexities.
• Carefully apply rules such as the distribution of powers over multiplication to tidy up expressions.

In our exercise,\[(2ab)^4\]applying the simplification process involves using properties of exponents. By simplifying \[2^4\]we calculate it as \[16\]and lay out \[a^4\]and\[b^4\]as individual terms.Ultimately, the simplified result \[16a^4b^4\]is a neater and more concise form of the original expression.

Power Rule

The power rule is a specific exponentiation guideline that tells us how to handle powers of products or numbers. It states that a power applies to all components in a base, \[(xy)^n = x^n y^n\]This means each part of the base is independently raised to a power.

• The rule ensures that each factor contributes its own share to the total power.
• This makes calculations systematic and reliable when dealing with algebraic expressions.

In our expression,\[(2ab)^4\]we can use the power rule to simplify it to \[2^4 a^4 b^4\]. After figuring that out, the rest is straightforward arithmetic and algebra, leading us to our final simplified expression,\[16a^4b^4\]. Grasping the power rule is essential for efficiently managing and simplifying algebraic expressions with multiple components.