Simplifying Algebraic Expressions A Comprehensive Guide

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Algebraic expressions, those fascinating combinations of variables, constants, and mathematical operations, can sometimes seem daunting in their complexity. But fear not, math enthusiasts! In this comprehensive guide, we'll break down the art of simplifying algebraic expressions, equipping you with the tools and techniques to conquer even the most intricate equations. Let's dive in and simplify the following expressions, making them more manageable and easier to understand.

a) Simplifying $\frac{-32 x^{10} y}{24 x^6 y^5}$

Let's start with the first expression: $\frac{-32 x^{10} y}{24 x^6 y^5}$. Our goal here is to reduce this fraction to its simplest form by canceling out common factors. First, we'll tackle the numerical coefficients, then we'll move on to the variables.

  • Simplifying Numerical Coefficients: Look at the coefficients -32 and 24. We need to find the greatest common divisor (GCD) of these numbers. The GCD of 32 and 24 is 8. So, we divide both the numerator and the denominator by 8.

    βˆ’32Γ·824Γ·8=βˆ’43\frac{-32 \div 8}{24 \div 8} = \frac{-4}{3}

  • Simplifying Variables: Now, let's deal with the variables. We have $x^10}$ in the numerator and $x^6$ in the denominator. When dividing variables with exponents, we subtract the exponents $x^{10 \div x^6 = x^10-6} = x^4$. Similarly, we have $y$ in the numerator (which is $y^1$) and $y^5$ in the denominator $y^1 \div y^5 = y^{1-5 = y^-4}$. Remember, a negative exponent means we can move the variable to the denominator to make the exponent positive $y^{-4 = \frac{1}{y^4}$.

  • Putting It All Together: Combining the simplified coefficients and variables, we get:

    βˆ’4x43y4\frac{-4 x^4}{3 y^4}

So, the simplified form of $\frac{-32 x^{10} y}{24 x^6 y^5}$ is $\frac{-4 x^4}{3 y^4}$. Remember, guys, always look for common factors in both numbers and variables – that’s the key to simplifying like a pro! This process of simplification not only makes expressions easier to handle but also reveals their underlying structure, which is crucial in solving more complex problems. The ability to quickly and accurately simplify algebraic fractions is a fundamental skill in algebra and calculus, and mastering it will undoubtedly boost your confidence in tackling mathematical challenges. Keep practicing, and you'll become a simplification superstar in no time!

b) Simplifying $\frac{6...$

Unfortunately, the expression in part (b) is incomplete. To provide a comprehensive explanation, I need the full expression. But hey, let's use this as an opportunity to talk about the general approach to simplifying more complex expressions! Simplifying algebraic expressions often involves a combination of several techniques, such as distributing, combining like terms, and factoring. When you encounter a complex expression, start by identifying the operations you can perform. Look for opportunities to distribute terms, combine like terms, or factor out common factors. Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). By following these steps, you can systematically simplify even the most intimidating expressions. This methodical approach will not only help you get the correct answer but also improve your understanding of the underlying mathematical principles. So, don't be afraid to break down complex problems into smaller, manageable steps. It's like building a house – you start with the foundation and work your way up, one step at a time. And just like in construction, precision and attention to detail are key to success. Make sure to double-check your work at each step to avoid errors, and don't hesitate to seek help or clarification if you're unsure about something. Math is a journey, and every problem you solve is a step forward on that journey!

e) Simplifying $\frac{-9 x^9+18 x^9}{-(-\left(3 x2\right)8)}$

Alright, let's tackle expression (e): $\frac{-9 x^9+18 x^9}{-(-\left(3 x2\right)8)}$. This one looks a bit intimidating, but don't worry, we'll break it down step by step. Remember, the key to simplifying algebraic expressions is to follow the order of operations and simplify each part systematically.

  • Simplifying the Numerator: First, let's simplify the numerator: $-9 x^9 + 18 x^9$. These are like terms (they have the same variable and exponent), so we can combine them: $-9 x^9 + 18 x^9 = (18 - 9) x^9 = 9 x^9$

  • Simplifying the Denominator: Now, let's simplify the denominator: $-(-\left(3 x2\right)8)$. We need to deal with the exponent first. Remember that when raising a product to a power, we raise each factor to that power:

    (3 x^2)^8 = 3^8 \cdot (x^2)^8$ Now, $3^8 = 6561$, and when raising a power to a power, we multiply the exponents: $(x^2)^8 = x^{2 \cdot 8} = x^{16}$. So, $(3 x^2)^8 = 6561 x^{16}$. Now, let's deal with the negative signs. We have $-\(-\left(6561 x^{16}\right))$. The two negative signs cancel each other out, so we're left with $6561 x^{16}$.

  • Putting It All Together: Now we have the simplified numerator and denominator, so we can write the simplified fraction:

    9x96561x16\frac{9 x^9}{6561 x^{16}}

  • Further Simplification: We can simplify this fraction further by dividing both the numerator and the denominator by their greatest common factor. The GCD of 9 and 6561 is 9, so we divide both by 9:

    9Γ·96561Γ·9=1729\frac{9 \div 9}{6561 \div 9} = \frac{1}{729}

    For the variables, we have $x^9$ in the numerator and $x^{16}$ in the denominator. Subtracting the exponents, we get $x^{9-16} = x^{-7} = \frac{1}{x^7}$.

  • Final Simplified Form: Combining the simplified coefficients and variables, we get:

    1729x7\frac{1}{729 x^7}

So, the simplified form of $\frac{-9 x^9+18 x^9}{-(-\left(3 x2\right)8)}$ is $\frac{1}{729 x^7}$. See, guys? It looks scary at first, but breaking it down into smaller steps makes it much more manageable. Always remember to simplify the numerator and denominator separately before combining them, and don't forget to look for opportunities to simplify further by canceling out common factors. This systematic approach is key to success in algebra and beyond. Keep practicing, and you'll become a simplification master!

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

Okay, let's dive into expression (i): $\frac{\left(-4 x^2 y^4\right)\left(-2 x^4 y^6\right)}{(-2 x y)^4} \div \frac{x y^6}{2}$. This one involves multiple operations, but we'll tackle it step by step. The key to simplifying algebraic expressions like this is to follow the order of operations (PEMDAS/BODMAS) and simplify each part systematically.

  • Simplifying the Numerator of the Main Fraction: First, let's simplify the numerator of the main fraction: $\left(-4 x^2 y^4\right)\left(-2 x^4 y^6\right)$. When multiplying terms with the same base, we multiply the coefficients and add the exponents:

    (βˆ’4)β‹…(βˆ’2)=8(-4) \cdot (-2) = 8

    x2β‹…x4=x2+4=x6x^2 \cdot x^4 = x^{2+4} = x^6

    y4β‹…y6=y4+6=y10y^4 \cdot y^6 = y^{4+6} = y^{10}

    So, the numerator simplifies to $8 x^6 y^{10}$.

  • Simplifying the Denominator of the Main Fraction: Next, let's simplify the denominator of the main fraction: $(-2 x y)^4$. We need to raise each factor inside the parentheses to the power of 4:

    (βˆ’2)4=16(-2)^4 = 16

    (x)4=x4(x)^4 = x^4

    (y)4=y4(y)^4 = y^4

    So, the denominator simplifies to $16 x^4 y^4$.

  • Simplifying the Main Fraction: Now we have the simplified numerator and denominator, so we can write the simplified main fraction:

    8x6y1016x4y4\frac{8 x^6 y^{10}}{16 x^4 y^4}

    We can simplify this fraction by dividing both the numerator and the denominator by their greatest common factor. The GCD of 8 and 16 is 8, so we divide both by 8:

    8Γ·816Γ·8=12\frac{8 \div 8}{16 \div 8} = \frac{1}{2}

    For the variables, we subtract the exponents: $x^{6-4} = x^2$ and $y^{10-4} = y^6$.

    So, the simplified main fraction is $\frac{x^2 y^6}{2}$.

  • Dealing with the Division: Now we have the expression $\frac{x^2 y^6}{2} \div \frac{x y^6}{2}$. Remember that dividing by a fraction is the same as multiplying by its reciprocal. So, we flip the second fraction and multiply:

    x2y62β‹…2xy6\frac{x^2 y^6}{2} \cdot \frac{2}{x y^6}

  • Simplifying the Result: Now we multiply the fractions:

    x2y6β‹…22β‹…xy6\frac{x^2 y^6 \cdot 2}{2 \cdot x y^6}

    We can cancel out the common factors: the 2s cancel out, $x^2 \div x = x$, and $y^6 \div y^6 = 1$.

  • Final Simplified Form: So, the simplified expression is $x$.

Wow, guys, we did it! The simplified form of $\frac{\left(-4 x^2 y^4\right)\left(-2 x^4 y^6\right)}{(-2 x y)^4} \div \frac{x y^6}{2}$ is simply $x$. This problem demonstrates the power of breaking down complex expressions into smaller, manageable steps. Remember to follow the order of operations, simplify each part systematically, and look for opportunities to cancel out common factors. With practice, you'll become a simplification wizard, effortlessly transforming complex expressions into their simplest forms.

In conclusion, simplifying algebraic expressions is a fundamental skill in mathematics. By mastering the techniques discussed in this guide, you'll be well-equipped to tackle a wide range of algebraic problems. Keep practicing, and remember that every simplified expression is a step closer to mathematical mastery!