Problem 50 - College Algebra with Modeling and Visualization

Question

Exercises \(45-54:\) Use the quotient rule to simplify the expression. Use positive exponents to write your answer. $$ \frac{10 x^{5}}{5 x^{-3}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 2x^{8} \).

Step 1: Identifying the Quotient Rule

The quotient rule in algebra states that when dividing two exponents of the same base, you can subtract the exponents. So, for an expression \( \frac{a^m}{a^n} \), it simplifies to \( a^{m-n} \).

Step 2: Apply the Quotient Rule to the Coefficients

Extract the coefficients from the expression, \( \frac{10}{5} \), which simplifies to \( 2 \). This is because dividing 10 by 5 results in 2.

Step 3: Apply the Quotient Rule to the Exponents

For the given expression \( \frac{x^5}{x^{-3}} \), apply the quotient rule to the exponents: subtract \( -3 \) from 5, which means \( 5 - (-3) = 5 + 3 = 8 \). So, the expression becomes \( x^8 \).

Step 4: Compile the Full Simplified Expression

Combine the simplified coefficient and the simplified variable expression: \( 2x^8 \). This is the simplified expression using positive exponents.

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of a number by itself. In mathematics, an exponent indicates how many times a base number is multiplied by itself. For example, in the expression \( x^5 \), the base is \( x \) and the exponent is 5, meaning \( x \times x \times x \times x \times x \).

Managing exponents correctly is crucial in simplifying expressions, especially when dealing with the same base. The primary rules include:

• Product Rule: When multiplying two exponents with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• Quotient Rule: When dividing, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power Rule: When raising an exponent to another power, you multiply exponents: \( (a^m)^n = a^{m \times n} \).

Understanding these rules allows for easy manipulation of expressions involving exponents in algebra.

Simplification

Simplification refers to the process of reducing an algebraic expression to its simplest form. This involves using mathematical rules to make an expression easier to understand and work with.

In the context of exponents, simplification often involves applying rules like the quotient rule, product rule, and others—especially when expressions include terms that share the same base.

For example, in the expression \( \frac{10 x^{5}}{5 x^{-3}} \), simplification involves:

• Dividing the coefficients: \( \frac{10}{5} = 2 \).
• Using the quotient rule for exponents: \( x^{5 - (-3)} = x^{5 + 3} = x^8 \).

Through simplification, the initial complex expression simplifies to a more manageable form, \( 2x^8 \), which is easier to use in calculations or further analysis.

Positive Exponents

Positive exponents are an essential concept in algebra and exponential mathematics. A positive exponent indicates straightforward multiplication of the base number, which contrasts with negative exponents that imply division or fractions.

When simplifying expressions, we aim to express all exponents positively for clarity and ease of understanding. This can be particularly useful in algebraic expressions or equations where clear and positive notation helps to avoid errors.

For instance, in simplifying the expression \( \frac{x^5}{x^{-3}} \), initially, this has a negative exponent in the denominator. Applying the quotient rule gives us positive exponents: \( x^{5 - (-3)} = x^{5 + 3} = x^8 \). The final simplified form of the expression is \( 2x^8 \), where both coefficients and exponents are positive, resulting in an expression that is easily interpreted and used.

Maintaining positive exponents is a crucial step in ensuring that the expressions are clear and straightforward, particularly when collaborating or communicating these results.