Simplifying Algebraic Expressions: A Step-by-Step Guide

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Hey guys! Today, we're diving into the fascinating world of algebra to tackle a common challenge: simplifying expressions. If you've ever felt lost in a maze of exponents and variables, don't worry! This guide will break down the process step-by-step, making it super easy to understand. We'll be focusing on simplifying three different algebraic expressions, each with its own unique twist. So, grab your pencils, and let's get started!

a) Simplifying 2a³ * (-3a⁴b⁵)³

Let's kick things off with our first expression: 2a³ * (-3a⁴b⁵)³. Simplifying algebraic expressions might seem daunting initially, but it's all about applying the right rules in the correct order. Our primary goal here is to reduce this expression to its simplest form by dealing with the exponents and coefficients. The main keywords here are simplifying expressions, algebraic expressions, and exponents. Understanding these concepts is crucial for mastering algebra.

Step-by-Step Breakdown

  1. Address the Power of a Product: The first hurdle is the term (-3a⁴b⁵)³. Remember, when you have a product raised to a power, you need to apply that power to each factor within the parentheses. This means we need to cube -3, a⁴, and b⁵ individually.

    • (-3)³ = -27 (because -3 * -3 * -3 = -27)
    • (a⁴)³ = a¹² (using the power of a power rule: (am)n = a^(m*n))
    • (b⁵)³ = b¹⁵ (again, using the power of a power rule)

    So, (-3a⁴b⁵)³ simplifies to -27a¹²b¹⁵.

  2. Rewrite the Expression: Now we can rewrite the original expression with this simplified term:

    2a³ * (-27a¹²b¹⁵)

  3. Multiply Coefficients: Next, we multiply the coefficients (the numerical parts) together:

    2 * -27 = -54

  4. Multiply Variables: Now, let's multiply the variables. We have a³ and a¹². When multiplying variables with the same base, you add the exponents:

    a³ * a¹² = a^(3+12) = a¹⁵

  5. Combine Terms: Finally, we combine everything together. We have the coefficient -54, the variable a raised to the power of 15 (a¹⁵), and b raised to the power of 15 (b¹⁵). So, the simplified expression is:

    -54a¹⁵b¹⁵

Key Takeaways

  • Power of a Product: Remember to distribute the exponent to each factor inside the parentheses.
  • Power of a Power: When raising a power to another power, multiply the exponents.
  • Multiplying Variables: When multiplying variables with the same base, add the exponents.

By following these steps, we've successfully simplified our first expression! You see, guys, it's all about breaking it down into manageable chunks. Understanding the fundamental rules of exponents and how they apply to different parts of the expression is key. Keep these principles in mind, and you'll be simplifying complex expressions like a pro in no time!

b) Simplifying (-x⁶y⁷)*(-xy¹⁰)²

Alright, let's move on to our next challenge: (-x⁶y⁷)*(-xy¹⁰)². This expression builds upon the skills we used in the first one, but it introduces another layer of complexity. We'll continue to focus on simplifying this expression using the order of operations and exponent rules. The core concepts here remain simplifying algebraic expressions and the application of exponent rules. Mastering these will significantly boost your algebra skills.

Step-by-Step Breakdown

  1. Address the Power of a Product: Just like before, we need to tackle the term raised to a power first, which is (-xy¹⁰)². We'll apply the power of 2 to both -x and y¹⁰.

    • (-x)² = x² (since (-1)² = 1 and x² = x²)
    • (y¹⁰)² = y²⁰ (using the power of a power rule: (ym)n = y^(m*n))

    So, (-xy¹⁰)² simplifies to x²y²⁰.

  2. Rewrite the Expression: Now, let's rewrite the original expression with our simplified term:

    (-x⁶y⁷) * (x²y²⁰)

  3. Multiply Variables: We're ready to multiply the variables together. Remember, when multiplying variables with the same base, we add their exponents.

    • x⁶ * x² = x^(6+2) = x⁸
    • y⁷ * y²⁰ = y^(7+20) = y²⁷

  4. Consider the Negative Sign: Don't forget about the negative sign in front of the first term! Since we're multiplying a negative term by a positive term, the result will be negative.

    -1 * x⁸y²⁷ = -x⁸y²⁷

  5. Combine Terms: Putting it all together, the simplified expression is:

    -x⁸y²⁷

Key Takeaways

  • Paying Attention to Signs: Always be mindful of negative signs, especially when squaring terms. A negative number squared becomes positive.
  • Consistency with Exponent Rules: Continue to apply the power of a power rule and the rule for multiplying variables with the same base.

See, guys? This one wasn't so bad either! The key is to take it one step at a time and focus on applying the rules consistently. By breaking down the expression and addressing each part methodically, you can simplify even the most intimidating-looking algebraic expressions. Practice makes perfect, so keep at it!

c) Simplifying 1⅘xy * (-1⅔x³y⁴)²

Last but not least, let's tackle the third expression: 1⅘xy * (-1⅔x³y⁴)². This one throws in a bit of a curveball with mixed numbers, but don't let that scare you! We'll convert those mixed numbers into improper fractions, making the calculations much easier. The main challenge here is dealing with mixed numbers within algebraic expressions, but the underlying principle of simplifying expressions remains the same.

Step-by-Step Breakdown

  1. Convert Mixed Numbers to Improper Fractions: Before we do anything else, we need to convert the mixed numbers into improper fractions.

    • 1⅘ = (1 * 5 + 4) / 5 = 9/5
    • -1⅔ = -(1 * 3 + 2) / 3 = -5/3

    So, our expression now looks like this:

    (9/5)xy * ((-5/3)x³y⁴)²

  2. Address the Power of a Product: Now, we'll handle the term raised to a power, which is ((-5/3)x³y⁴)². We apply the power of 2 to each factor inside the parentheses.

    • (-5/3)² = (-5/3) * (-5/3) = 25/9
    • (x³)² = x⁶ (using the power of a power rule)
    • (y⁴)² = y⁸ (using the power of a power rule)

    So, ((-5/3)x³y⁴)² simplifies to (25/9)x⁶y⁸.

  3. Rewrite the Expression: Let's rewrite the original expression with this simplified term:

    (9/5)xy * (25/9)x⁶y⁸

  4. Multiply Coefficients: Now, we multiply the coefficients (the fractions) together:

    (9/5) * (25/9) = (9 * 25) / (5 * 9) = 225/45

    We can simplify this fraction by dividing both numerator and denominator by their greatest common divisor, which is 45:

    225/45 = 5

  5. Multiply Variables: Next, we multiply the variables together:

    • x * x⁶ = x^(1+6) = x⁷
    • y * y⁸ = y^(1+8) = y⁹

  6. Combine Terms: Finally, we combine the simplified coefficient and variables:

    5x⁷y⁹

Key Takeaways

  • Converting Mixed Numbers: Always convert mixed numbers to improper fractions before performing calculations. It makes things much smoother!
  • Fraction Multiplication: Remember the rules for multiplying fractions: multiply the numerators and multiply the denominators.
  • Simplifying Fractions: Don't forget to simplify your fractions whenever possible.

Awesome job, guys! We've conquered our third expression, even with those tricky mixed numbers. This example highlights the importance of having a solid foundation in basic arithmetic as well as algebraic rules. By converting to improper fractions and applying the same exponent rules, we were able to simplify this expression just as effectively as the previous ones.

Conclusion

So, there you have it! We've successfully simplified three different algebraic expressions, each with its own unique challenges. Remember, guys, the key to simplifying algebraic expressions is to break them down into smaller, manageable steps. By following the order of operations, applying exponent rules correctly, and paying close attention to signs and coefficients, you can tackle even the most complex expressions with confidence. Keep practicing, and you'll become an algebra whiz in no time! I hope this guide has been helpful, and remember, don't be afraid to ask for help when you need it. Happy simplifying! Understanding algebraic simplification is crucial for success in higher-level math, so keep honing those skills!**