Simplifying The Algebraic Expression (5y^3)(-6y^7)(2y^2) A Comprehensive Guide

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In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. It makes complex problems more manageable and understandable. In this comprehensive guide, we'll delve into the process of simplifying the expression extbf{${ (5y3)(-6y7)(2y^2) }$}, breaking it down step-by-step to ensure clarity. This article aims to provide a detailed explanation suitable for students and anyone looking to enhance their understanding of algebraic manipulation.

Understanding Algebraic Expressions

Before we dive into the simplification process, let's first understand what algebraic expressions are and the key components involved. Algebraic expressions are combinations of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Variables are symbols (usually letters) that represent unknown values, while constants are fixed numerical values. Understanding the structure of these expressions is crucial for simplification.

Key Components of Algebraic Expressions

  1.  extbf{Variables:} These are symbols, typically letters like `x`, `y`, or `z`, that represent unknown quantities. In our expression, `y` is the variable.

  2.  extbf{Constants:} These are fixed numerical values. In our expression, we have constants such as `5`, `-6`, and `2`.

  3.  extbf{Coefficients:} These are the numerical values multiplied by the variables. For instance, in the term `5y^3`, `5` is the coefficient.

  4.  extbf{Exponents:} These indicate the power to which a variable or constant is raised. In the term `y^3`, `3` is the exponent.

  5.  extbf{Terms:} These are the individual parts of the expression separated by addition or subtraction. In our case, we have three terms implicitly multiplied together: `5y^3`, `-6y^7`, and `2y^2`.

Basic Rules of Algebra

To simplify algebraic expressions effectively, it's essential to be familiar with some basic rules of algebra. These include:

  •   extbf{Commutative Property:} This property states that the order of operations does not affect the result for addition and multiplication (e.g., \${ a + b = b + a \}$ and \${ a \\times b = b \\times a \}$).

  •   extbf{Associative Property:} This property states that the grouping of terms does not affect the result for addition and multiplication (e.g., \${ (a + b) + c = a + (b + c) \}$ and \${ (a \\times b) \\times c = a \\times (b \\times c) \}$).

  •   extbf{Distributive Property:} This property allows us to multiply a single term by multiple terms inside parentheses (e.g., \${ a(b + c) = ab + ac \}$).

  •   extbf{Exponent Rules:} These rules govern how to handle exponents in algebraic expressions. We'll delve into these in more detail later.

Step-by-Step Simplification of extbf{${ (5y3)(-6y7)(2y^2) }$}

Now, let's break down the simplification of the expression extbf{${ (5y3)(-6y7)(2y^2) }$} step-by-step. The key to simplifying this expression lies in understanding how to multiply terms with the same base and how to apply the rules of exponents.

Step 1: Multiply the Coefficients

The first step in simplifying the expression is to multiply the coefficients. The coefficients in our expression are 5, -6, and 2. Multiplying these together gives us:

${ 5 \times -6 \times 2 = -60 }$

So, the numerical part of our simplified expression will be -60.

Step 2: Multiply the Variables

Next, we need to multiply the variable terms, which are y^3, y^7, and y^2. When multiplying terms with the same base, we add their exponents. This is a fundamental rule of exponents:

${ a^m \times a^n = a^{m+n} }$

Applying this rule to our variables, we get:

${ y^3 \times y^7 \times y^2 = y^{3+7+2} = y^{12} }$

Thus, the variable part of our simplified expression is y^{12}.

Step 3: Combine the Results

Now that we have simplified both the coefficients and the variables, we can combine them to get the final simplified expression. We multiply the numerical coefficient -60 by the variable term y^{12}:

${ -60 \times y^{12} = -60y^{12} }$

Therefore, the simplified form of the expression extbf{${ (5y3)(-6y7)(2y^2) }} is extbf{\${-60y^{12}\}}.

Detailed Explanation of Exponent Rules

As we saw in Step 2, the exponent rules play a crucial role in simplifying algebraic expressions. Let's delve deeper into these rules to ensure a solid understanding.

Product of Powers Rule

This rule, which we used in our simplification, states that when multiplying powers with the same base, you add the exponents:

${ a^m \times a^n = a^{m+n} }$

This rule is essential for simplifying expressions involving multiplication of variables with exponents. For example:

  • ${ x^2 \times x^3 = x^{2+3} = x^5 }$
  • ${ 2y^4 \times 3y^2 = 2 \times 3 \times y^{4+2} = 6y^6 }$

Power of a Power Rule

This rule states that when raising a power to another power, you multiply the exponents:

${ (am)n = a^{m \times n} }$

For example:

  • ${ (x3)2 = x^{3 \times 2} = x^6 }$
  • ${ (y2)5 = y^{2 \times 5} = y^{10} }$

Power of a Product Rule

This rule states that when raising a product to a power, you raise each factor to that power:

${ (ab)^n = a^n b^n }$

For example:

  • ${ (2x)^3 = 2^3 x^3 = 8x^3 }$
  • ${ (3y2)2 = 3^2 (y2)2 = 9y^4 }$

Quotient of Powers Rule

This rule states that when dividing powers with the same base, you subtract the exponents:

${ \frac{am}{an} = a^{m-n} }$

For example:

  • ${ \frac{x5}{x2} = x^{5-2} = x^3 }$
  • ${ \frac{y7}{y3} = y^{7-3} = y^4 }$

Zero Exponent Rule

This rule states that any non-zero number raised to the power of zero is equal to 1:

${ a^0 = 1 }$ (where ${ a \neq 0 }$)

For example:

  • ${ 5^0 = 1 }$
  • ${ x^0 = 1 }$ (if ${ x \neq 0 }$)

Negative Exponent Rule

This rule states that a number raised to a negative exponent is equal to the reciprocal of the number raised to the positive exponent:

${ a^{-n} = \frac{1}{a^n} }$

For example:

  • ${ x^{-2} = \frac{1}{x^2} }$
  • ${ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} }$

Common Mistakes to Avoid

While simplifying algebraic expressions, it's easy to make mistakes if you're not careful. Here are some common pitfalls to avoid:

  1.  extbf{Incorrectly Applying Exponent Rules:} One of the most common mistakes is misapplying the exponent rules. For instance, confusing the product of powers rule with the power of a power rule can lead to incorrect simplifications. Always double-check which rule applies to the situation.

  2.  extbf{Forgetting to Multiply Coefficients:} When multiplying terms, it's crucial to multiply the coefficients as well as the variables. Overlooking the coefficients can result in an incorrect final answer.

  3.  extbf{Adding Exponents When Multiplying:} Remember that when multiplying terms with the same base, you add the exponents, not multiply them. For example, \${ x^2 \\times x^3 = x^5 \}$, not \${ x^6 \}$.

  4.  extbf{Ignoring the Order of Operations:} Always follow the order of operations (PEMDAS/BODMAS) to ensure correct simplification. Parentheses, Exponents, Multiplication and Division, Addition and Subtraction should be done in that order.

  5.  extbf{Not Simplifying Completely:} Make sure to simplify the expression as much as possible. This might involve combining like terms or further applying exponent rules.

Practice Problems

To solidify your understanding, let's go through a few more practice problems. These examples will help you apply the concepts we've discussed and improve your skills in simplifying algebraic expressions.

Practice Problem 1

Simplify the expression: extbf{${ (3x2y)(4xy3)(2x^3) }$}

extbf{Solution:}
  1. Multiply the coefficients: ${ 3 \times 4 \times 2 = 24 }$
  2. Multiply the x terms: ${ x^2 \times x \times x^3 = x^{2+1+3} = x^6 }$
  3. Multiply the y terms: ${ y \times y^3 = y^{1+3} = y^4 }$
  4. Combine the results: extbf{${\bf{24x6y4}}$}

Practice Problem 2

Simplify the expression: extbf{${ (-2a3b2)^3 }$}

extbf{Solution:}
  1. Apply the power of a product rule: ${ (-2)^3 (a3)3 (b2)3 }$
  2. Simplify each term: \
    • ${ (-2)^3 = -8 }$
    • ${ (a3)3 = a^{3 \times 3} = a^9 }$
    • ${ (b2)3 = b^{2 \times 3} = b^6 }$
  3. Combine the results: extbf{${\bf{-8a9b6}}$}

Practice Problem 3

Simplify the expression: extbf{${ \frac{15m5n2}{3m^2n} }$}

extbf{Solution:}
  1. Divide the coefficients: ${ \frac{15}{3} = 5 }$
  2. Divide the m terms: ${ \frac{m5}{m2} = m^{5-2} = m^3 }$
  3. Divide the n terms: ${ \frac{n^2}{n} = n^{2-1} = n }$
  4. Combine the results: extbf{${\bf{5m^3n}}$}

Advanced Techniques for Simplifying Expressions

Beyond the basic rules and steps, there are advanced techniques that can further simplify complex algebraic expressions. These techniques often involve recognizing patterns, factoring, and using special algebraic identities.

Factoring

Factoring is the process of breaking down an expression into its constituent factors. This can be particularly useful when simplifying expressions involving polynomials. Common factoring techniques include:

  •   extbf{Greatest Common Factor (GCF):} Identifying and factoring out the GCF from all terms in the expression.

  •   extbf{Difference of Squares:} Recognizing and factoring expressions in the form \${ a^2 - b^2 \}$ as \${ (a + b)(a - b) \}$.

  •   extbf{Perfect Square Trinomials:} Recognizing and factoring expressions in the form \${ a^2 + 2ab + b^2 \}$ as \${ (a + b)^2 \}$ or \${ a^2 - 2ab + b^2 \}$ as \${ (a - b)^2 \}$.

  •   extbf{Factoring by Grouping:} Grouping terms in pairs to find common factors.

Algebraic Identities

Algebraic identities are equations that are always true, regardless of the values of the variables. These identities can be used to simplify expressions more efficiently. Some common algebraic identities include:

  •   extbf{\${(a + b)^2 = a^2 + 2ab + b^2\}$}

  •   extbf{\${(a - b)^2 = a^2 - 2ab + b^2\}$}

  •   extbf{\${(a + b)(a - b) = a^2 - b^2\}$}

  •   extbf{\${(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\}$}

  •   extbf{\${(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\}$}

  •   extbf{\${(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\}$}

  •   extbf{\${(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\}$}

Complex Expressions

When dealing with more complex expressions, it's essential to break the problem down into smaller, more manageable steps. This might involve a combination of the techniques we've discussed, such as applying exponent rules, factoring, and using algebraic identities. Patience and careful attention to detail are key to successfully simplifying these expressions.

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics. By understanding the basic rules, exponent rules, and advanced techniques such as factoring and algebraic identities, you can tackle even the most complex expressions. Remember to practice regularly and pay attention to detail to avoid common mistakes. With a solid understanding and consistent practice, you'll become proficient in simplifying algebraic expressions and build a strong foundation for further mathematical studies.

Simplifying the expression extbf{${ (5y3)(-6y7)(2y^2) }} involves multiplying the coefficients and adding the exponents of the variables. By following the step-by-step process, we found that the simplified form is extbf{\${-60y^{12}\}}. This guide has provided a thorough explanation and additional practice problems to help you master this essential skill. Keep practicing, and you'll find that simplifying algebraic expressions becomes second nature.