Problem 63 - 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. $$ \frac{x^{5} y^{7}}{x^{3} y^{4}} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: $\frac{x^{5} y^{7}}{x^{3} y^{4}}$ Answer: $x^{2}y^{3}$

Step 1: Apply the Quotient Rule for Exponents

To start simplifying the expression, we can apply the Quotient Rule which states that: $$ \frac{a^{m}}{a^{n}} = a^{m-n} $$ Applying this to our expression, we get: $$ \frac{x^{5} y^{7}}{x^{3} y^{4}}=\frac{x^{5}}{x^{3}}\cdot\frac{y^{7}}{y^{4}} $$

Step 2: Simplify the expression using the Quotient Rule

Now, we can simplify both the x and y terms individually using the Quotient Rule: $$ \frac{x^{5}}{x^{3}}=x^{5-3}=x^{2} \quad \text{and} \quad \frac{y^{7}}{y^{4}}=y^{7-4}=y^{3} $$

Step 3: Combine the simplified terms using the Product Rule

Lastly, we can now combine the simplified x and y terms using the Product Rule, which states that: $$ a^{m} \cdot a^{n} = a^{m+n} $$ For our expression, we have: $$ x^{2} \cdot y^{3} $$ However, since x and y are different bases, we cannot use the Product Rule here, and our final expression is already in its simplest form. So, the simplified form of the given expression is: $$ \frac{x^{5} y^{7}}{x^{3} y^{4}} = x^{2}y^{3} $$

Key Concepts

Product Rule of Exponents

In the world of algebra, understanding the product rule of exponents is essential for simplifying expressions efficiently. The product rule states that when you multiply like bases, you add their exponents. For example, if you have expressions like \(a^m \cdot a^n\), you can simplify this to \(a^{m+n}\).
Applying this in algebraic problems helps streamline the solving process and reduces complex expressions into more manageable forms. While using this rule, always make sure that the bases are identical. If the bases differ, the product rule does not apply.

Quotient Rule of Exponents

The quotient rule of exponents is another handy tool in simplifying expressions where both the numerator and denominator have the same base. This rule states: \(\frac{a^m}{a^n} = a^{m-n}\).
This means that when dividing like bases, you simply subtract the exponent in the denominator from the exponent in the numerator.
In the example given, \(\frac{x^5 y^7}{x^3 y^4}\), the quotient rule is used on each variable separately. For the x terms, \(x^5\) divided by \(x^3\) results in \(x^{5-3} = x^2\). Likewise, for the y terms, \(y^7\) divided by \(y^4\) simplifies to \(y^{7-4} = y^3\).
This process ensures that complex algebraic fractions become simple and easy to work with.

Simplifying Expressions

Simplifying expressions involves breaking down complex algebraic statements into their simplest form. This process often involves applying various algebraic rules, such as the product and quotient rules of exponents.
To simplify an expression, always ensure that you handle each section of the expression methodically, often by breaking it into smaller parts. This allows for a clearer understanding of how each part interacts.
In our example, after applying the quotient rule, we were left with the simplified terms \(x^2\) and \(y^3\).
By simplifying accurately, mistakes can be minimized, and algebra problems become more approachable and less intimidating, making them easier to solve.

Whole Number Exponents

Whole number exponents refer to exponents that are positive integers or zero. They indicate how many times the base is multiplied by itself.
For example, \(x^3\) means \(x\) is multiplied three times: \(x \cdot x \cdot x\). Whole number exponents come with straightforward rules that are easy to understand and apply, such as the product rule (adding exponents) and quotient rule (subtracting exponents).
Understanding how to efficiently use whole number exponents is crucial in simplifying and solving algebraic expressions. It lays the groundwork for dealing with more complex exponent rules and makes handling algebraic functions smoother. Proper application helps maintain the integrity of expressions while transforming them.