Problem 45 - Elementary and Intermediate Algebra

Question

Use the quotient rule for exponents to simplify each expression. Write the results using exponents. $$ \frac{c^{3} d^{7}}{c d} $$

Step-by-Step Solution

Answer

\( c^2 d^6 \)

Step 1: Apply the Quotient Rule

The quotient rule of exponents states that for any non-zero number \( a \), \( \frac{a^m}{a^n} = a^{m-n} \). We'll apply this rule separately to each variable. Start by applying it to the variable \( c \).

Step 2: Simplify the Expression for \( c \)

The expression for \( c \) in the numerator is \( c^3 \) and in the denominator is \( c \), which is equivalent to \( c^1 \). Apply the quotient rule: \[ \frac{c^3}{c^1} = c^{3-1} = c^2 \].

Step 3: Simplify the Expression for \( d \)

The expression for \( d \) in the numerator is \( d^7 \) and in the denominator is \( d \), which is also \( d^1 \). Apply the quotient rule: \[ \frac{d^7}{d^1} = d^{7-1} = d^6 \].

Step 4: Combine the Simplified Terms

Now that we have simplified both parts, combine them back into one expression: \( c^2 d^6 \). This is the simplified form using exponents.

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in algebra that helps make an equation easier to solve or understand. When faced with complex algebraic expressions, simplification reduces the complexity by reducing the number of operations needed. This step often involves eliminating common factors, combining similar terms, or in the case of exponential expressions, using rules like the quotient rule. In our given problem, we have two algebraic expressions in the numerator and denominator, each containing variables with exponents. By applying the quotient rule for exponents, we simplify the expression

• Break down each variable into parts you can manage independently,
• Simplify these parts and
• Combine them back to achieve a concise expression.

Exponents

Exponents represent repeated multiplication of the same number. They are often seen in expressions like \( a^n \), where \( a \) is the base and \( n \) is the exponent. Instead of writing out multiple multiplications, exponents condense this information into a single term. Some key points about exponents include:

• An expression like \( a^3 \) means \( a \times a \times a \).
• The quotient rule for exponents, \( \frac{a^m}{a^n} = a^{m-n} \), lets you divide powers of the same base by subtracting their exponents. This can only be applied if the base is not zero.

In our exercise, we applied this rule to simplify both \( c^3/c \) and \( d^7/d \). By doing this, we managed to reduce the expression down to \( c^2d^6 \). It is crucial when simplifying that you always check your initial and simplified forms to ensure they represent the same quantity.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. These expressions are foundational in algebra as they are the building blocks for equations and functions. Learning to work with algebraic expressions is crucial for developing strong mathematical skills. Here are some important aspects of algebraic expressions:

• Variables in expressions like \( c \) and \( d \) can represent unknown values or numbers that vary.
• Operations (addition, subtraction, multiplication, division) are used to combine numbers and variables in diverse ways.
• Simplifying them helps in solving algebraic equations or inequalities more efficiently because it reduces the complexity involved.

In the context of our exercise, we encountered the algebraic expression \( \frac{c^{3} d^{7}}{c d} \). By understanding the roles of exponents and the quotient rule, we were able to break it down systematically. This resulted in \( c^2d^6 \), a simpler form that's easier to work with for further calculations or analysis.