Problem 58 - 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. $$ \frac{y^{5}}{y^{2}} $$

Step-by-Step Solution

Answer

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

Step 1: Recall the Quotient Rule of Exponents

The quotient rule of exponents states that for any nonzero base b and whole number exponents m and n: $$ \frac{b^{m}}{b^{n}} = b^{m-n} $$ This rule is useful for simplifying expressions involving division of exponentials with the same base.

Step 2: Apply the Quotient Rule to the given expression

We are given the expression: $$ \frac{y^{5}}{y^{2}} $$ Since the base is the same (y), we can directly apply the quotient rule of exponents: $$ \frac{y^{5}}{y^{2}} = y^{5-2} $$

Step 3: Calculate the final exponent

Now, subtract the exponents according to the rule: $$ y^{5-2} = y^{3} $$

Step 4: Write the simplified expression

The simplified expression after applying the quotient rule of exponents is: $$ y^3 $$

Key Concepts

Product Rule of Exponents

Whenever you encounter expressions that involve multiplying two or more terms with the same base, the product rule of exponents is your go-to tool. The rule states that for any nonzero base \( b \) and whole number exponents \( m \) and \( n \): \[ b^m \times b^n = b^{m+n} \]This simply means that when you multiply, you add the exponents if the bases are the same. Let's examine why this works:
* Each exponent tells you how many times to multiply the base by itself. * So, when you multiply the same bases together, you are essentially adding how many times the base is used in the multiplication.Imagine multiplying \( a^2 \times a^3 \). According to the product rule, you get:\[ a^2 \times a^3 = a^{2+3} = a^5 \]You have five \( a \)s being multiplied altogether. This concept is crucial whenever you simplify algebraic expressions, making your calculations faster and easier!

Quotient Rule of Exponents

The quotient rule of exponents is an ideal method to simplify expressions involving division with the same base. The rule articulates that for any nonzero base \( b \) and whole number exponents \( m \) and \( n \): \[ \frac{b^m}{b^n} = b^{m-n} \]Essentially, you subtract the exponent of the denominator from the exponent of the numerator. Here’s the breakdown as to why it’s effective:

• When dividing, you are identifying the difference in the number of times the base is multiplied on top versus the bottom.
• This ensures only the net multiplications of the base remain after simplification.

Consider the expression \( \frac{y^5}{y^2} \). If you apply the quotient rule, you rewrite it as:\[ y^{5-2} = y^3 \]You’ve effectively removed the factor of \( y^2 \) in the numerator, leaving a cleaner simplified form \( y^3 \). This rule is particularly valuable as it prevents divide-and-conquer errors, and it swiftly shows how the power balances after division.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra, allowing you to express mathematical statements in their simplest form. This often involves using specific rules, such as the product and quotient rules of exponents, to make expressions easier to work with.
Here are the foundational steps to simplify expressions involving exponents:

• Identify common bases in your terms and examine what operation (multiplication or division) you are performing.
• Utilize exponent rules like the product or quotient rule appropriately to combine or cancel out exponents.
• Always perform arithmetic on the exponents after determining the rule to apply.

For example, the expression \( \frac{y^5}{y^2} \) simplifies efficiently through the quotient rule by getting:\[ y^{5-2} = y^3 \]By consistently applying such rules with care, you ensure accurate and simplified results, leaving you with an equation that’s more straightforward to analyze or use in further calculations. Remember, simplifying makes life easier!