Problem 43 - Elementary and Intermediate Algebra

Question

Use the quotient rule for exponents to simplify each expression. Write the results using exponents. $$ \frac{(3.7 p)^{7}}{(3.7 p)^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \((3.7p)^5\).

Step 1: Identify the Base and Exponents

The expression \( \frac{(3.7 p)^{7}}{(3.7 p)^{2}} \) has a common base \( 3.7p \) both in the numerator and the denominator. The exponents are 7 and 2 respectively.

Step 2: Apply the Quotient Rule for Exponents

The quotient rule for exponents states that \( \frac{a^m}{a^n} = a^{m-n} \), given the same base \(a\). Apply this rule here to get: \( (3.7p)^{7-2} \).

Step 3: Simplify the Expression

Subtract the exponents: \( 7 - 2 = 5 \). So, the expression simplifies to \( (3.7p)^{5} \).

Key Concepts

Quotient Rule for Exponents

The quotient rule is an essential tool when dealing with expressions that have powers or exponents. If you have the same base in both the numerator and denominator of a fraction, this rule simplifies the expression.

In essence, the rule is:

• For any non-zero number \(a\) and integers \(m\) and \(n\), the rule states \(\frac{a^m}{a^n} = a^{m-n}\).

Here, the base \(a\) is common in both the numerator and denominator, which allows us to subtract the exponents to simplify. This is particularly powerful when working with large expressions, as it reduces the complexity.

The key takeaway is that the quotient rule only works when the expressions share the same base. If the bases differ, this rule cannot be applied. So in any problem, always look to see if the exponents share a common base before using this rule.

Simplifying Expressions

Simplifying expressions with exponents involves making expressions as concise as possible, reducing them to their simplest form. This process often includes the application of rules such as the quotient rule.

In the exercise provided, simplifying takes place over several steps:

• First, identify the common base in the expression.
• Use rules like the quotient rule to manipulate the exponents.

For instance, given an expression \(\frac{(3.7 p)^7}{(3.7 p)^2}\), recognizing \(3.7p\) as the common base allows you to apply the quotient rule. After subtraction of the exponents (\(7 - 2\)), you simplify it to \((3.7p)^5\).

Ensuring expressions are concise helps in solving more complex problems and keeps calculations straightforward. It's important to go through each step carefully to avoid errors, especially with subtraction in the exponents.

Base and Exponents

Understanding what constitutes a base and an exponent is foundational in working with expressions involving powers.

• The **base** is the number or variable that is repeatedly multiplied.
• The **exponent** indicates how many times the base is used in the multiplication.

For example, in \(3^4\), '3' is the base, and '4' is the exponent, meaning \(3\) is multiplied by itself 4 times leading to \(3 \times 3 \times 3 \times 3\).

In algebraic expressions like \((3.7p)^7\), the base becomes everything inside the parentheses, here, \(3.7p\).

Knowing these basics helps in simplifying expressions, especially when using rules like the quotient rule. By focusing on the base and understanding how to manipulate exponents, students can tackle more complex algebraic concepts with confidence.