Problem 47 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(\frac{-4 d^{-1}}{p^{-10}}\)

Step-by-Step Solution

Answer

The simplified expression is \( -\frac{4}{dp^{10}} \).

Step 1: Understand the Problem

We have the expression \( \frac{-4 d^{-1}}{p^{-10}} \) that needs to be simplified with no negative exponents in the final answer. Recall that a negative exponent \( x^{-a} \) is equivalent to \( \frac{1}{x^a} \).

Step 2: Simplify Negative Exponents in the Numerator

The term in the numerator \(-4d^{-1}\) can be rewritten using the property of negative exponents: \(-4d^{-1} = \frac{-4}{d^1} = \frac{-4}{d}\).

Step 3: Simplify Negative Exponents in the Denominator

Similarly, apply the property of negative exponents to the denominator \(p^{-10}\): it becomes \(\frac{1}{p^{-10}} = p^{10}\). Thus, the denominator becomes a positive exponent term \(p^{10}\).

Step 4: Combine and Simplify the Expression

Substitute the simplified numerator and denominator into the original fraction: \(\frac{\frac{-4}{d}}{p^{10}} = -4 \cdot \frac{1}{d} \cdot \frac{1}{p^{10}}\). This can be rewritten as: \(-\frac{4}{d \cdot p^{10}}\).

Step 5: Final Simplified Expression

The simplified expression of \( \frac{-4 d^{-1}}{p^{-10}} \) with no negative exponents is \( -\frac{4}{dp^{10}} \).

Key Concepts

Negative Exponents

Negative exponents can seem a bit tricky at first, but they're easier to handle once you get the hang of them. Any number raised to a negative exponent is essentially indicating the reciprocal of that number raised to the corresponding positive exponent.
For example, if you see \( x^{-a} \), it means \( \frac{1}{x^a} \).

• Think of a negative exponent as simply an instruction to "flip" the base (put it under a 1 if it's not already in a fraction).
• So, for \( d^{-1} \), it becomes \( \frac{1}{d} \).
• Similarly, \( p^{-10} \) would turn into \( \frac{1}{p^{10}} \).

It's good to remember this rule because it helps you rewrite expressions without negative exponents, which is often required in algebra to ensure simplicity and clarity.

Fraction Simplification

Simplifying fractions involves reducing them to their simplest form by eliminating complex expressions or finding common factors. When dealing with expression fractions that include variable exponents, you often have to manage both numbers and variables.
Here's the general process:

• Simplify the top (numerator) and the bottom (denominator) separately. Apply rules for operations on both parts.
• Cancel common factors from both the numerator and denominator. Remember, this can include variables with exponents as well.
• Simplify the expression by applying arithmetic operations like multiplication or division where needed.

In our example \( \frac{-4 d^{-1}}{p^{-10}} \), the simplified expression becomes \( -\frac{4}{d \cdot p^{10}} \) after applying these rules. Notice how handling negative exponents was part of simplifying the fraction.

Negative to Positive Exponents

Converting negative exponents to positive is a critical skill in algebra. Having expressions with positive exponents is often preferred because they are easier to interpret and calculate.
Here’s how you go about it:

• Identify the terms with negative exponents.
• Rewrite these terms as their reciprocals with positive exponents. For instance, changing \( a^{-b} \) into \( \frac{1}{a^b} \).
• In complex fractions, keep applying these steps until no negative exponents remain.

Using this method helps in keeping mathematical expressions neat.For instance, the \( p^{-10} \) in our equation was changed into \( p^{10} \) when moved from the denominator to the numerator, demonstrating how seamlessly this transition can be applied. The final expression \( -\frac{4}{dp^{10}} \) is an excellent example of managing and simplifying negative exponents effectively.