Problem 48 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(\frac{-9 m^{-1}}{n^{-30}}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-9n^{30}}{m} \).

Step 1: Identify the expression

The expression given is \( \frac{-9 m^{-1}}{n^{-30}} \). Our goal is to simplify this expression while ensuring all exponents are positive.

Step 2: Convert negative exponents to positive exponents

To convert negative exponents to positive, move the term with a negative exponent across the fraction bar. Move \( m^{-1} \) from the numerator to the denominator and \( n^{-30} \) from the denominator to the numerator. The expression becomes \( -9 \times n^{30} \times m^{1} \), which simplifies to \( \frac{-9n^{30}}{m} \).

Step 3: Write the simplified expression

After simplification, the expression is \( \frac{-9n^{30}}{m} \). This expression has no negative exponents, meeting the requirement of the problem.

Key Concepts

Negative Exponents

Negative exponents can be thought of as the opposite of positive exponents. Let's explore what this means. In mathematics, an exponent indicates how many times a number, or 'base', is multiplied by itself. For example, in the term \(3^4\), the number 3 is multiplied by itself 4 times: \(3 \times 3 \times 3 \times 3\).
Now, what happens if the exponent is negative? A negative exponent tells us to do the opposite: instead of multiplying, we divide the base. So, \(3^{-4}\) is the same as \(\frac{1}{3^4}\), which means \(\frac{1}{3 \times 3 \times 3 \times 3}\).
In general, a negative exponent \(a^{-n}\) means \(\frac{1}{a^n}\), where \(a\) is the base and \(n\) is the exponent. Remember to flip the fraction or reciprocal the base to change the sign of the exponent from negative to positive.

Fraction Simplification

Simplifying fractions can make expressions easier to work with. It involves reducing fractions to their simplest form by eliminating common factors and dealing with exponents appropriately.
Here's a quick guide to simplifying fractions:

• Look for common factors in the numerator and denominator. If there's any, divide them out.
• What if you have terms with exponents? Reduce or eliminate any negative exponents first by moving terms across the fraction line.
• Once exponents are positive and common factors are canceled, you'll have a simplified fraction.

In our example with \(\frac{-9 m^{-1}}{n^{-30}}\), moving terms to make exponents positive and then rewriting them gives us \(\frac{-9n^{30}}{m}\).
This process is essential, especially when expressions become complex. Simplifying fractions helps in identifying core components of an expression and preparing it for further computation.

Positive Exponents

Positive exponents are the straightforward way we indicate multiplication of a base. When an exponent is positive, it tells you how many times to multiply the base by itself.
For example, with \(x^5\), you multiply \(x\) by itself 5 times: \(x \times x \times x \times x \times x\). For expressions, positive exponents are generally preferred when simplifying because they make calculations and expressions much cleaner.
Consider positive exponents as more user-friendly when performing algebraic manipulations. In our simplified expression \(\frac{-9n^{30}}{m}\), the exponent \(30\) on \(n\) is positive, which signifies that you should multiply \(n\) by itself 30 times.
Working with positive exponents is crucial for avoiding the confusion that negative exponents might bring. Always look to write expressions without negatives when possible, just like we did by changing \(n^{-30}\) to \(n^{30}\). This highlights a cleaner and clearer view of the expression's structure.