Problem 27 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ (3 a b)^{4} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((3ab)^4\). Answer: \(81a^4b^4\)

Step 1: Identify the base and exponent

In the given expression, \((3ab)^4\), the base is \(3ab\) and the exponent is \(4\).

Step 2: Apply the power rule for exponents

According to the power rule for exponents, \((ab)^{n} = a^{n} \times b^{n}\), where \(a\) and \(b\) are the bases and \(n\) is the exponent. Apply this rule to the given expression: \((3ab)^4 = 3^4 \times a^4 \times b^4\).

Step 3: Calculate the constant term

Calculate the value for the constant term \(3^4\): \(3^4 = 3\times3\times3\times3 = 81\).

Step 4: Write the final result

The simplified expression is: \(81a^4b^4\).

Key Concepts

Power Rule for Exponents

When dealing with exponents, one of the most important rules is the power rule. This rule is applied when you have an entire expression raised to a power. The power rule states that

• when you have \((a \, b)^n\), it becomes \(a^n \times b^n\).
• You distribute the exponent outside the parentheses to each element inside.

This is very helpful when simplifying complex algebraic expressions involving exponents.
In simpler terms, if you encounter something like \((x \times y)^3\), you would use the power rule to expand it as \(x^3 \times y^3\).
This method ensures that each base inside the parentheses is raised individually to the given power before any multiplication takes place.
Using this rule simplifies calculations and makes it easier to manage and manipulate algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They can contain:

• numbers (known as constants),
• variables (often represented by letters such as \(a, b, x\), or \(y\)), and
• operations (addition, subtraction, multiplication, and division).

In the given exercise, the expression \((3ab)^4\) is an algebraic expression.
This particular form consists of a constant number \(3\) and variables \(a\) and \(b\), all raised collectively to the power of 4.
Generally, simplifying algebraic expressions involves combining like terms and applying laws of arithmetic and exponents to reduce the complexity of the expression.
The goal is to make them simpler to work with, which often involves eliminating parentheses using distribution rules, like the power rule, ensuring each component is simplified appropriately.

Natural Numbers

Natural numbers are the simplest set of numbers that include all positive integers starting from 1. They are represented as:

• \(1, 2, 3, 4, 5, ...\)

These numbers are fundamental in counting and ordering, and in the given exercise, natural numbers play a vital role in the exponents.
Exponents in our example are a great instance of applying natural numbers, since the exponent \(4\) signifies that the base must be multiplied by itself 4 times.
This concept is crucial because it tells us exactly how many times the base participates in the multiplication, which helps in breaking down and simplifying expressions like \((3ab)^4\).
Knowing that exponents are natural numbers allows us to deduce straightforward calculations like \(3^4 = 81\), using a repetition of basic multiplication.