Problem 60 - Algebra A Combined Function

Question

Simplify each expression. Write each result using positive exponents only. $$ \frac{-15 r^{-6} s}{5 r^{-4} s^{-3}} $$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{3s^4}{r^2}\).

Step 1: Simplify the coefficients

First, simplify the fraction of the coefficients \(-15/5\). Divide \(-15\) by \(5\) to get \(-3\). This will become the new coefficient of the simplified expression.

Step 2: Apply the quotient rule for exponents

The quotient rule for exponents tells us that for any nonzero number \(a\), \( \frac{a^m}{a^n} = a^{m-n} \). Apply this rule to the variables. For \(r\), we have \(r^{-6}\) over \(r^{-4}\), which simplifies to \(r^{-6 - (-4)} = r^{-2}\). For \(s\), we have \(s^1\) over \(s^{-3}\), which simplifies to \(s^{1 - (-3)} = s^4\).

Step 3: Combine the results

Combine the simplified coefficient and the simplified variables to get the expression \(-3r^{-2}s^4\).

Step 4: Convert negative exponents to positive

To convert negative exponents to positive exponents, we use the rule \(a^{-m} = \frac{1}{a^m}\). So, \(r^{-2}\) becomes \(\frac{1}{r^2}\). Therefore, the expression becomes \(-3 \cdot \frac{s^4}{r^2}\) which simplifies to \(-\frac{3s^4}{r^2}\).

Key Concepts

Exponents

Exponents are a foundational concept in algebra. They are used to express how many times a number, known as the base, is multiplied by itself. For instance, the expression \(a^m\) means that the base \(a\) is used as a factor \(m\) times. This concept can significantly simplify complex calculations, making it easier to handle large numbers. For example:

• \(2^3 = 2 \times 2 \times 2 = 8\)
• \(10^2 = 10 \times 10 = 100\)

By using exponents, we can quickly understand and manipulate large numbers without having to write them out completely. This concept is crucial for simplifying algebraic expressions and solving real-world problems efficiently.

Quotient Rule for Exponents

The quotient rule for exponents is a useful tool when dividing expressions with the same base. This rule states that for any nonzero base number \(a\), and any integers \(m\) and \(n\), the expression \(\frac{a^m}{a^n}\) simplifies to \(a^{m-n}\). This is because when dividing exponential terms with the same base, you subtract the exponent of the denominator from the exponent of the numerator.
For example, let's say you have the expression \(\frac{x^5}{x^3}\). Using the quotient rule:

• Subtract the exponent of the denominator from the exponent of the numerator: \(5 - 3 = 2\)
• The simplified expression is \(x^2\).

This rule simplifies the process of working with fractions that have the same base, and it's essential for simplifying expressions in algebra.

Negative Exponents

Negative exponents may seem tricky at first, but they follow a straightforward rule: any base raised to a negative exponent is equal to the reciprocal of the base raised to the opposite positive exponent. In mathematical terms, \(a^{-m} = \frac{1}{a^m}\). This tells us that negative exponents indicate division, rather than multiplication.
For instance:

• \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\)
• \(10^{-2} = \frac{1}{10^2} = \frac{1}{100}\)

Understanding and applying negative exponents is crucial when simplifying algebraic expressions, as it allows you to write expressions with positive exponents, which are typically easier to understand and work with. In the context of the problem, this rule was used to convert \(r^{-2}\) to \(\frac{1}{r^2}\), simplifying the expression to have all positive exponents.