Mastering Algebraic Simplification: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving deep into the world of algebraic simplification. This stuff is super important because it forms the foundation for more advanced math concepts. Whether you're just starting out or need a refresher, this guide is for you. We will break down each problem into manageable steps, ensuring you grasp the core concepts. Ready to simplify some expressions? Let's get started!

Simplifying Expressions: A Comprehensive Approach

(a) Simplifying $\frac{\left(4 x y^{-2}\right)\left(-12 x^{-4} y^{12}\right)}{6 x^2 y}$

Alright, let's kick things off with our first problem, which involves simplifying a fraction containing variables and exponents. This might look a little intimidating at first, but trust me, it's totally manageable. The key is to break it down into smaller, easier-to-handle parts. We'll use the properties of exponents and basic arithmetic to arrive at a simplified answer. First things first, we'll deal with the numerator. We have two terms multiplied together: (4xy⁻²) and (-12x⁻⁴y¹²). When multiplying these terms, remember to multiply the coefficients (the numbers in front) and combine the variables using the exponent rules. The rule we need here is: xᵃ * xᵇ = xᵃ⁺ᵇ. Essentially, when multiplying terms with the same base (like x), you add the exponents.

So, let's work on the numerator: (4xy⁻²) * (-12x⁻⁴y¹²) = (4 * -12) * (x¹ * x⁻⁴) * (y⁻² * y¹²). Now, let's simplify each part. 4 * -12 equals -48. For the x terms, x¹ * x⁻⁴ becomes x¹⁻⁴ = x⁻³. For the y terms, y⁻² * y¹² becomes y⁻²+¹² = y¹⁰. Putting it all together, the numerator simplifies to -48x⁻³y¹⁰. Now we can rewrite the original expression with the simplified numerator, so we have: -48x⁻³y¹⁰ / (6x²y). Next, we'll divide this by the denominator, which is 6x²y. Divide the coefficient -48 by 6 which gives us -8. For the x terms, we have x⁻³ / x². Using the rule xᵃ / xᵇ = xᵃ⁻ᵇ, we get x⁻³⁻² = x⁻⁵. For the y terms, we have y¹⁰ / y¹. This simplifies to y¹⁰⁻¹ = y⁹. Our expression now becomes -8x⁻⁵y⁹. One last thing, it is generally preferred to have positive exponents in our final answer. To convert x⁻⁵ to a positive exponent, we can move it to the denominator (remember, x⁻ᵃ = 1/xᵃ). Therefore, -8x⁻⁵y⁹ becomes -8y⁹/x⁵. Voila! We've successfully simplified the expression to -8y⁹/x⁵. It might seem like a lot of steps, but with practice, you'll get the hang of it. Remember to take it step by step, and don't be afraid to write out all the details.

Key Steps:

• Multiply the terms in the numerator. Remember to multiply coefficients and add exponents of the same variables.
• Simplify the numerical part of the fraction. Divide the coefficients.
• Simplify the variable parts. Use the rule xᵃ / xᵇ = xᵃ⁻ᵇ.
• Ensure all exponents are positive. Move any terms with negative exponents to the opposite side of the fraction.

(b) Simplifying $\left(2 x^{-1} y^{-2}\right)^{-3}\left(4 x^2 y^3\right)^4$

Now, let's tackle the second expression: (2x⁻¹y⁻²)⁻³(4x²y³)⁴. This one introduces a new twist: the power of a power. The rule to remember here is (xᵃ)ᵇ = xᵃᵇ. Basically, when you have a power raised to another power, you multiply the exponents. We'll start by dealing with each set of parentheses separately. First, we'll focus on (2x⁻¹y⁻²)⁻³. We'll distribute the -3 to each term inside the parentheses. So we get: 2⁻³ * (x⁻¹)⁻³ * (y⁻²)⁻³. Let's simplify each part. 2⁻³ is 1/2³ = 1/8. (x⁻¹)⁻³ becomes x⁻¹⁻³ = x³. (y⁻²)⁻³ becomes y⁻²*⁻³ = y⁶. Putting it all together, the first part simplifies to (1/8)x³y⁶.

Next, let's simplify (4x²y³)⁴. Distribute the 4 to each term inside the parentheses: 4⁴ * (x²)⁴ * (y³)⁴. Simplify each part. 4⁴ = 256. (x²)⁴ becomes x²⁴ = x⁸. (y³)⁴ becomes y³⁴ = y¹². So, the second part simplifies to 256x⁸y¹². Now we have: (1/8)x³y⁶ * 256x⁸y¹². Multiply the coefficients: (1/8) * 256 = 32. For the x terms, we have x³ * x⁸ which becomes x³⁺⁸ = x¹¹. For the y terms, we have y⁶ * y¹² which becomes y⁶⁺¹² = y¹⁸. Putting everything together, our final answer is 32x¹¹y¹⁸. See? Breaking it down step by step makes it so much easier.

Key Steps:

• Apply the power of a product rule. Distribute the outer exponent to each term inside the parentheses.
• Simplify each term. Calculate numerical values and multiply exponents.
• Combine like terms. Multiply coefficients and add exponents of like variables.

(c) Simplifying $\sqrt[2]{\left(9 x^6 y^{-2} z^4\right)^3}(3 x y z)^{-2}$

Alright, let's dive into the final problem: √((9x⁶y⁻²z⁴)³) / (3xyz)². This one involves both radicals and negative exponents, so we'll need to apply all the rules we've learned so far! First, let's simplify inside the square root. We have (9x⁶y⁻²z⁴)³. Distribute the exponent 3 to each term: 9³ * (x⁶)³ * (y⁻²)³ * (z⁴)³. Calculate each part. 9³ = 729. (x⁶)³ becomes x⁶³ = x¹⁸. (y⁻²)³ becomes y⁻²³ = y⁻⁶. (z⁴)³ becomes z⁴*³ = z¹². So, the expression inside the square root simplifies to 729x¹⁸y⁻⁶z¹². Now we have: √(729x¹⁸y⁻⁶z¹²). The square root of 729 is 27, so we have 27√(x¹⁸y⁻⁶z¹²). For the variables, we take the square root of each term which means dividing the exponents by 2: √(x¹⁸) = x⁹, √(y⁻⁶) = y⁻³, √(z¹²) = z⁶.

Therefore, √(729x¹⁸y⁻⁶z¹²) = 27x⁹y⁻³z⁶. Now, the entire expression becomes: 27x⁹y⁻³z⁶ / (3xyz)². Now let's deal with the second part, (3xyz)⁻². When you have a negative exponent, you can move the entire term to the denominator and make the exponent positive, or distribute the exponent to each term as we did before. Let's apply it that way and simplify each of them, (3xyz)⁻² becomes 3⁻² * x⁻² * y⁻² * z⁻². Calculate each term. 3⁻² is 1/3² = 1/9. Now the entire expression becomes: (27x⁹y⁻³z⁶) * (1/9x²y²z²). Then, multiply the fraction, which is 27/9 = 3. For the x terms, we have x⁹ / x² = x⁹⁻² = x⁷. For the y terms, we have y⁻³ / y² = y⁻³⁻² = y⁻⁵. For the z terms, we have z⁶ / z² = z⁶⁻² = z⁴. So we get 3x⁷y⁻⁵z⁴. Since we want to avoid negative exponents, we rewrite it with the y⁻⁵ moved to the denominator: 3x⁷z⁴/y⁵. And that's the simplified form. You did it!

Key Steps:

• Simplify the terms inside the radical. Apply the power of a product rule.
• Simplify the radical. Take the square root of the numerical and variable parts.
• Simplify the term with the negative exponent. Apply the power of a product rule.
• Combine like terms and simplify the expression.
• Ensure all exponents are positive. Move any terms with negative exponents to the opposite side of the fraction.

Conclusion: Mastering Algebraic Simplification

Congratulations, you made it through all the problems! We've covered the core principles of simplifying algebraic expressions, including working with exponents, fractions, and radicals. Remember, practice is key! The more you work through problems, the more comfortable you'll become with these concepts. Don't get discouraged if it seems tough at first. Keep practicing, break down each problem into smaller parts, and you'll become a simplification master in no time. Keep up the great work, and happy simplifying!