Problem 53 - 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. $$ \frac{x^{5}}{x^{3}} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression using the product rule and quotient rule of exponents: \(\frac{x^{5}}{x^{3}}\). Answer: The simplified expression is \(x^{2}\).

Step 1: Recall the Quotient Rule of Exponents

According to the quotient rule of exponents, when you divide two expressions with the same base and the exponents are whole numbers, you can subtract the exponent in the denominator from the exponent in the numerator. In other words, \(\frac{a^{m}}{a^{n}} = a^{m-n}\).

Step 2: Apply the Quotient Rule to the Given Expression

Using the quotient rule, we can simplify the given expression, \(\frac{x^{5}}{x^{3}}\): $$ \frac{x^{5}}{x^{3}} = x^{5-3}. $$

Step 3: Simplify the Exponent

Now, we need to subtract the exponents: $$ x^{5-3} = x^{2}. $$

Step 4: Final Answer

The simplified expression is: $$ x^{2}. $$

Key Concepts

Product Rule

The product rule is a fundamental concept when dealing with powers in algebra. It’s a handy tool used when you multiply two powers that have the same base. Imagine you have two expressions like \( a^m \) and \( a^n \). The product rule tells us that we can multiply these by simply adding their exponents:

• $$ a^{m} \times a^{n} = a^{m+n} $$

The process simplifies working with exponents significantly since you don't need to expand each expression fully, saving time and reducing mistakes. Remember that the bases must be the same to apply the product rule. If they aren't the same, this rule doesn't apply.

Exponents

Exponents are a way of expressing repeated multiplication of the same number. They allow us to write long multiplication expressions in a compact way.For example, the expression \( x^5 \) means that the base \( x \) is multiplied by itself 5 times:

• $$ x^5 = x \times x \times x \times x \times x $$

Exponents play a key role in many mathematical operations, and their properties, like the product and quotient rules, help simplify calculations a lot. They are used in various mathematical fields and also in real-life scenarios involving growth, like computing compound interest or population growth. The notation \( a^n \) comes with a base \( a \) and an exponent \( n \). It’s important to recognize and correctly apply operations to exponents to make the most of their power in simplifying expressions.

Simplification

Simplification is the process of transforming an expression into its simplest form. This might mean reducing the number of terms, or changing the form to make it more understandable or easier to work with. In algebra, you often simplify by using rules like the product rule or the quotient rule to combine or reduce terms. Take the expression \( \frac{x^5}{x^3} \) for example. By applying the quotient rule of exponents, which involves subtracting the exponent in the denominator from the exponent in the numerator, you simplify it to \( x^{5-3} = x^2 \). Simplification helps make complex expressions easier to understand and solve, and it’s key to solving equations in algebra. It’s all about working smarter, not harder.

Algebraic Expressions

Algebraic expressions are groups of symbols that can include numbers, variables, and operations. They are like sentences in the language of mathematics. An algebraic expression can be as simple as a single number or variable, like \( x \), or more complex, like \( 2x + 3y - 5 \). Understanding algebraic expressions involves knowing how to work with each component, including variables (which are symbols like \( x \) or \( y \) that represent numbers) and constants (which are specific numbers). They can also include operations like addition, subtraction, multiplication, and division.To work with algebraic expressions, you must utilize rules and operations such as the product rule, quotient rule, and various algebra techniques to manipulate and solve them. Being familiar with these helps you to simplify or evaluate expressions, making problem-solving much easier.