Simplify Expression: Find The Exponent Of B

Let's dive into simplifying algebraic expressions! In this article, we'll break down how to correctly simplify the expression $\frac{-4 a^{-2} b^4}{8 a^{-6} b^{-3}}$. We'll focus especially on finding the correct exponent for the variable b, which Marina needs a little help with. So, grab your algebraic tools, and let's get started!

Understanding the Problem

Algebraic expressions can sometimes look intimidating, especially with negative exponents and fractions involved. The key is to remember the rules of exponents and how to apply them correctly. We're given the expression $\frac{-4 a^{-2} b^4}{8 a^{-6} b^{-3}}$ and Marina's partially simplified form: $-\frac{1}{2} a^4 b^{\square}$. Our mission is to find the missing exponent in Marina's solution. This involves simplifying the given expression and comparing it with Marina's result. Remember, a solid grasp of exponent rules, like the quotient rule and the negative exponent rule, is super important to navigate through these problems effectively. So, let's roll up our sleeves and get down to the nitty-gritty of simplifying this expression and unveiling the mystery exponent of b.

Step-by-Step Simplification

To correctly simplify the given expression, $\frac{-4 a^{-2} b^4}{8 a^{-6} b^{-3}}$, we'll go step by step, focusing on each part of the expression:

1. Simplify the coefficients:

   • We have $\frac{-4}{8}$, which simplifies to $-\frac{1}{2}$.

2. Simplify the variable a:

   • We have $\frac{a^{-2}}{a^{-6}}$. Using the quotient rule ($\frac{a^m}{a^n} = a^{m-n}$), we get $a^{-2 - (-6)} = a^{-2 + 6} = a^4$.

3. Simplify the variable b:

   • We have $\frac{b^4}{b^{-3}}$. Using the quotient rule again, we get $b^{4 - (-3)} = b^{4 + 3} = b^7$.

Putting it all together, the simplified expression is $-\frac{1}{2} a^4 b^7$. This methodical approach helps in breaking down complex expressions into manageable parts, making it easier to apply the necessary rules and arrive at the correct simplified form. Each step is crucial, from simplifying coefficients to applying the quotient rule for variables with exponents.

Finding the Exponent of b

Now that we've simplified the expression to $-\frac{1}{2} a^4 b^7$, we can easily identify the exponent of b. Comparing this with Marina's expression, $-\frac{1}{2} a^4 b^{\square}$, it's clear that the missing exponent is 7. Therefore, the correct exponent of the variable b in Marina's solution should be 7. This precise comparison not only completes Marina's simplified expression but also confirms the accuracy of our step-by-step simplification process. This exercise highlights the significance of understanding and applying exponent rules correctly to achieve accurate results in algebraic simplifications. By carefully working through each component of the expression, we successfully pinpointed the exponent of b, ensuring Marina's solution is complete and correct.

Common Mistakes to Avoid

When simplifying expressions with exponents, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them:

• Incorrectly applying the quotient rule: Remember that when dividing terms with the same base, you subtract the exponents. For example, $\frac{a^5}{a^2} = a^{5-2} = a^3$, not $a^{5/2}$.
• Forgetting the negative sign: When dealing with negative coefficients, make sure to carry the negative sign through the simplification. For instance, $\frac{-4}{8}$ simplifies to $-\frac{1}{2}$, not $\frac{1}{2}$.
• Misunderstanding negative exponents: A negative exponent means you take the reciprocal of the base raised to the positive exponent. For example, $a^{-3} = \frac{1}{a^3}$. When simplifying fractions with negative exponents, remember to move the term to the opposite side of the fraction.
• Adding exponents when multiplying: When multiplying terms with the same base, you add the exponents. For example, $a^2 * a^3 = a^{2+3} = a^5$. Many students mistakenly multiply the exponents instead.

By being mindful of these common errors and practicing regularly, you can improve your accuracy and confidence in simplifying algebraic expressions. Always double-check your work and pay close attention to the signs and exponent rules.

Practice Problems

To solidify your understanding of simplifying expressions with exponents, here are a few practice problems:

1. Simplify: $\frac{9x^{-3}y^2}{3x^2y^{-5}}$
2. Simplify: $\frac{-12a^4b^{-1}}{4a^{-2}b^3}$
3. Simplify: $\frac{15c^{-5}d^{-3}}{-5c^{-1}d^2}$

Try solving these problems on your own, and then check your answers with the solutions provided below:

1. Solution: $\frac{9x^{-3}y^2}{3x^2y^{-5}} = 3x^{-5}y^7 = \frac{3y^7}{x^5}$
2. Solution: $\frac{-12a^4b^{-1}}{4a^{-2}b^3} = -3a^6b^{-4} = \frac{-3a^6}{b^4}$
3. Solution: $\frac{15c^{-5}d^{-3}}{-5c^{-1}d^2} = -3c^{-4}d^{-5} = \frac{-3}{c^4d^5}$

Working through these practice problems will not only reinforce your understanding but also help you identify any areas where you may need further clarification. Consistent practice is key to mastering algebraic simplifications.

Conclusion

In conclusion, simplifying algebraic expressions involves a clear understanding of exponent rules and careful attention to detail. By following a systematic approach, we were able to correctly simplify the expression $\frac{-4 a^{-2} b^4}{8 a^{-6} b^{-3}}$ and determine that the exponent of b in Marina's simplified expression should be 7. Avoiding common mistakes and practicing regularly will help you build confidence and accuracy in simplifying algebraic expressions. So keep practicing, and you'll become a pro at simplifying expressions in no time! Remember, algebra is a journey, and every problem you solve brings you one step closer to mastery. Keep up the great work!