Problem 75 - Elementary Algebra

Question

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ \frac{(x+3 y)^{11}(2 x-1)^{4}}{(x+3 y)^{3}(2 x-1)} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\frac{(x+3 y)^{11}(2 x-1)^{4}}{(x+3 y)^{3}(2 x-1)}\). Answer: \((x+3y)^8(2x-1)^3\)

Step 1: Identify the rules to use

We are given an expression with powers of sums, and we need to simplify it using the product rule and the quotient rule of exponents. The product rule states that a^(m+n) = a^m * a^n and the quotient rule states that a^(m-n) = a^m / a^n.

Step 2: Apply the quotient rule

Let's apply the quotient rule to the factor \((x+3y)\). We have \((x+3y)^{11}\) in the numerator and \((x+3y)^3\) in the denominator, so applying the quotient rule, we get: \((x+3y)^{11-3} = (x+3y)^8\) Now, let's apply the quotient rule to the factor \((2x-1)\). We have \((2x-1)^4\) in the numerator and \((2x-1)\) in the denominator. Applying the quotient rule, we get: \((2x-1)^{4-1} = (2x-1)^3\)

Step 3: Combine the results

Now that we have simplified each part, let's combine the results to obtain the final simplified expression: $$ \frac{(x+3 y)^{11}(2 x-1)^{4}}{(x+3 y)^{3}(2 x-1)} = (x+3y)^8(2x-1)^3 $$ So, the simplified expression is \((x+3y)^8(2x-1)^3\).

Key Concepts

Product Rule

The product rule is a handy tool when you're dealing with expressions that have the same base raised to different exponents. It's a method to simplify or expand these expressions easily. When you multiply two powers with the same base, you simply add the exponents together.
For example, if you have \(a^m \times a^n\), you can rewrite it as \(a^{m+n}\). This rule applies because multiplying like bases essentially asks how many times you need to multiply the base by itself.

• Useful for multiplying terms with the same base.
• Simplifies the multiplication process by reducing several steps into one simple addition.

In the context of our exercise, although we didn't directly apply the product rule since the expression was structured with division, understanding it helps in grasping the power manipulation concepts better.

Quotient Rule

The quotient rule is crucial when simplifying expressions where a base with an exponent in the numerator appears with a similar base in the denominator. It simplifies such cases by subtracting the exponents.
The rule states that if you have \(a^m \div a^n\), it can be rewritten as \(a^{m-n}\). This simplification arises because division is essentially the inverse operation of multiplication.

• Essential for dealing with fractional exponents.
• Helps reduce complex expressions into more manageable terms.

In our example, the quotient rule helped us simplify \((x+3y)^{11} \div (x+3y)^3\) to \((x+3y)^8\) by subtracting the exponents 11 and 3.

Simplifying Expressions

Simplifying expressions in algebra often involves applying rules like the product and quotient rules to make expressions shorter and easier to understand. Simplification makes it easier to solve equations and can make further operations more straightforward.
In the given exercise, the expression \(\frac{(x+3 y)^{11}(2 x-1)^{4}}{(x+3 y)^{3}(2 x-1)}\) is made simpler by strategically applying the quotient rule twice.

• The powers of like terms are reduced by subtracting their exponents.
• Creates a cleaner and more efficient expression \((x+3y)^8(2x-1)^3\).

By simplifying, we not only make it easier to work with the expression but also make it possible to apply it in different contexts or plug in numerical values easily.

Whole Numbers

Whole numbers are the set of numbers that include all positive integers beginning from zero. They do not include fractions, negatives, or decimals. When dealing with exponents, whole numbers are particularly simple because they lead to straightforward operations.
In our exercise context, we assume that all exponents are whole numbers, which means the operations are straightforward and don't involve fractions or negative results. This assumption fosters clarity and ease of computation.

• Simplifies calculations in algebraic expressions since fractions or decimals are not involved.
• Allows for predictable outcomes in exponentiation.

Understanding that the expression only includes whole numbers makes it easier to predict and verify the final simplified results, like in our example.