Simplify $\frac{8 a^3 b^{-1}}{2 a^4}$.

Question

06-11-2026
Mathematics

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Simplify $\frac{8 a^3 b^{-1}}{2 a^4}$.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-06T14:00:38-05:00

Simplify the constant coefficients by calculating $\frac{8}{2} = 4$.
Apply the quotient rule of exponents to the $a$ terms: $\frac{a^3}{a^4} = a^{3-4} = a^{-1}$.
Convert the negative exponent $b^{-1}$ to its fractional form $\frac{1}{b}$.
Combine all components to reach the final simplified form $\boxed{\frac{4}{ab}}$.

Explanation

Problem Analysis

We are given the algebraic expression $\frac{8 a^3 b^{-1}}{2 a^4}$. To simplify this, we need to apply the properties of exponents and basic arithmetic. We assume $a eq 0$ and $b eq 0$ for the expression to be well-defined. The expression consists of a numerator $8 a^3 b^{-1}$ and a denominator $2 a^4$. Our goal is to consolidate these terms into a simpler form.

Simplification Process

First, we simplify the constant coefficients. Dividing the numerator coefficient by the denominator coefficient gives:
\[ \frac{8}{2} = 4 \]
Next, we apply the quotient rule for exponents, which states that $\frac{x^m}{x^n} = x^{m-n}$, to the terms involving $a$:
\[ \frac{a^3}{a^4} = a^{3-4} = a^{-1} \]
Finally, we address the term $b^{-1}$. Using the definition of negative exponents, $b^{-1} = \frac{1}{b}$. Combining these parts, we get:
\[ 4 \cdot a^{-1} \cdot b^{-1} = 4 \cdot \frac{1}{a} \cdot \frac{1}{b} = \frac{4}{ab} \]
Thus, the simplified expression is $\frac{4}{ab}$.

Final Conclusion

By consolidating the coefficients and applying the laws of exponents, we have simplified the expression $\frac{8 a^3 b^{-1}}{2 a^4}$ to $\frac{4}{ab}$. This represents the simplest form of the given expression, assuming $a, b eq 0$. The final result is $\boxed{\frac{4}{ab}}$.

Examples

Algebraic simplification is a foundational skill used in many practical areas. For instance, if you are designing a computer program that calculates the cost of materials for a project, you might have complex formulas involving variables like the quantity of material ($a$) and a scaling factor ($b$). Simplifying these expressions before putting them into your code helps your program run faster and makes it easier for other engineers to read and maintain your code. Just as we reduced $\frac{8 a^3 b^{-1}}{2 a^4}$ to $\frac{4}{ab}$, you can often optimize your logical processes in daily tasks to save time and effort.