Problem 29 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. \(a b^{2} \div d^{-3}\)

Step-by-Step Solution

Answer

The simplified expression is \( a b^2 d^3 \).

Step 1: Understand the expression

The given expression is \( a b^2 \div d^{-3} \). This expression involves division of powers and the use of negative exponents.

Step 2: Apply the division of powers rule

When dividing powers, we effectively subtract the exponents of like bases. Here, we're dividing by \( d^{-3} \), which means we have \( d^{-3} \) in the denominator.

Step 3: Simplify using negative exponents

Convert the division by replacing the negative exponent \( d^{-3} \) for multiplication with positive exponent, which means multiplying by \( d^3 \). Thus, the expression becomes \( a b^2 imes d^3 \).

Step 4: Write the simplified expression

After converting the expression, the resulting simplified expression is \( a b^2 d^3 \). All exponents are now positive.

Key Concepts

Division of Powers Rule

The division of powers rule is a handy tool when working with exponents. It helps to slim down expressions by simplifying them. The basic premise is that when you divide powers that share the same base, you subtract the exponents. For example, if you have \( x^m \div x^n \), you can rewrite it as \( x^{m-n} \).
This rule streamlines complex expressions and makes them easier to manage. In our example, since \( d^{-3} \) is in the denominator and the expression requires division, applying this rule means subtracting the exponent in the denominator from the numerator. This operation simplifies the expression effectively by working with the exponents directly.

Negative Exponents

Negative exponents might seem tricky at first, but they are simply a way to represent fractions with positive exponents. When you have a negative exponent, like \( x^{-n} \), it's the same as the reciprocal of the positive exponent: \( \frac{1}{x^n} \). This means that negative exponents in the denominator can be flipped to the numerator as positive exponents.
Let's see this in practice. If you have \( x^{-3} \), it is equivalent to \( \frac{1}{x^3} \). Conversely, dividing by \( d^{-3} \) is the same as multiplying by \( d^3 \) when simplifying expressions. This move to positive exponents keeps things straightforward and is crucial when writing answers with only positive exponents.

Positive Exponents

Using positive exponents is often required in final answers for clarity. Positive exponents indicate that the number should be multiplied by itself a certain number of times. For instance, \( x^3 = x \times x \times x \).
When simplifying expressions, converting to positive exponents reduces complexity and maintains consistency. By turning any negative exponents into positive ones, we avoid unnecessary fractions in our solutions.

• Converting a negative to a positive exponent involves shifting the base from denominator to numerator or vice versa.
• Positive exponents simplify mathematical expressions, making them cleaner and easier to understand.

Keeping terms with positive exponents is standard practice in algebra, as in the case of simplifying \( a b^2 d^{-(-3)} = a b^2 d^3 \), resulting in a neat and clear expression.