Expanding Algebraic Expressions A Comprehensive Guide

In the realm of mathematics, expanding algebraic expressions is a fundamental skill. This article aims to provide a detailed exploration of expanding various algebraic expressions, focusing on examples labeled k through o. We will break down each expression, providing step-by-step solutions and explanations to enhance understanding. Mastery of these techniques is crucial for success in algebra and related fields, as it forms the basis for more advanced mathematical operations and problem-solving strategies. The ability to expand expressions accurately and efficiently not only simplifies calculations but also provides deeper insights into the structure and properties of algebraic equations. Through this guide, you will learn how to approach different types of expressions, apply relevant formulas, and simplify the results to their most basic forms. This comprehensive approach ensures a solid foundation in algebraic manipulation, preparing you for more complex mathematical challenges.

Expanding Expression k: $(\frac{2}{3} a^{6} b^{4} c^{-3} + 11 a b^{2})^{2}$

To expand the algebraic expression $(\frac{2}{3} a^{6} b^{4} c^{-3} + 11 a b^{2})^{2}$, we will employ the formula $(A + B)^{2} = A^{2} + 2AB + B^{2}$. This formula is a cornerstone of algebraic expansion and provides a structured approach to solving such problems. In this case, we identify $A$ as $\frac{2}{3} a^{6} b^{4} c^{-3}$ and $B$ as $11 a b^{2}$. By correctly identifying these components, we set the stage for a methodical expansion process that ensures accuracy and clarity. The application of this formula is not merely a mechanical process; it requires a clear understanding of algebraic principles and the ability to recognize patterns within the expression. This step-by-step approach not only facilitates the expansion but also enhances comprehension, making it easier to tackle similar problems in the future.

Step 1: Identify A and B

Let $A = \frac{2}{3} a^{6} b^{4} c^{-3}$ and $B = 11 a b^{2}$. This initial step is crucial as it sets the foundation for applying the expansion formula. Correctly identifying the components of the expression ensures that the subsequent calculations are accurate. It's important to pay close attention to the coefficients, variables, and their respective exponents, as any error in this stage will propagate through the rest of the solution. This meticulous approach helps in avoiding common mistakes and fosters a deeper understanding of the algebraic structure of the expression.

Step 2: Apply the Formula $(A + B)^{2} = A^{2} + 2AB + B^{2}$

Now, we substitute $A$ and $B$ into the formula: $(\frac{2}{3} a^{6} b^{4} c^{-3} + 11 a b^{2})^{2} = (\frac{2}{3} a^{6} b^{4} c^{-3})^{2} + 2(\frac{2}{3} a^{6} b^{4} c^{-3})(11 a b^{2}) + (11 a b^{2})^{2}$. This step involves careful substitution and attention to detail to ensure that all terms are correctly placed within the expansion. The expansion formula provides a systematic way to distribute the square over the sum of two terms, and its correct application is essential for obtaining the correct result. This process highlights the importance of algebraic manipulation skills and the ability to follow a structured approach in problem-solving.

Step 3: Calculate $A^{2}$

$(\frac{2}{3} a^{6} b^{4} c^{-3})^{2} = (\frac{2}{3})^{2} (a^{6})^{2} (b^{4})^{2} (c^{-3})^{2} = \frac{4}{9} a^{12} b^{8} c^{-6}$. This step involves applying the power of a product rule, which states that $(ab)^n = a^n b^n$, and the power of a power rule, which states that $(a^m)^n = a^{mn}$. By correctly applying these rules, we can simplify the squared term and prepare it for further calculations. Attention to the exponents and coefficients is crucial to avoid errors. This process demonstrates the importance of understanding and applying fundamental exponent rules in algebraic simplification.

Step 4: Calculate $2AB$

$2(\frac{2}{3} a^{6} b^{4} c^{-3})(11 a b^{2}) = 2 \cdot \frac{2}{3} \cdot 11 \cdot a^{6} \cdot a \cdot b^{4} \cdot b^{2} \cdot c^{-3} = \frac{44}{3} a^{7} b^{6} c^{-3}$. In this step, we multiply the coefficients and combine like terms by adding their exponents. The product rule for exponents, $a^m \cdot a^n = a^{m+n}$, is applied here to simplify the variable terms. Accurate multiplication and careful tracking of exponents are essential to ensure the correctness of the result. This process reinforces the importance of algebraic manipulation skills and the ability to combine like terms effectively.

Step 5: Calculate $B^{2}$

$(11 a b^{2})^{2} = 11^{2} (a)^{2} (b^{2})^{2} = 121 a^{2} b^{4}$. Similar to the calculation of $A^2$, this step involves applying the power of a product rule and the power of a power rule. Squaring the coefficient and applying the power rule to the variables results in the simplified term. This step further demonstrates the application of exponent rules and their role in algebraic simplification.

Step 6: Combine All Terms

Putting it all together, we have: $(\frac{2}{3} a^{6} b^{4} c^{-3} + 11 a b^{2})^{2} = \frac{4}{9} a^{12} b^{8} c^{-6} + \frac{44}{3} a^{7} b^{6} c^{-3} + 121 a^{2} b^{4}$. This final step involves combining the results from the previous steps to form the expanded expression. The terms are arranged in a logical order, typically by decreasing powers of the variables. This final expression represents the complete expansion of the original expression and showcases the application of algebraic principles and manipulation skills.

Expanding Expression l: $(5 x^{2} - 3)^{3}$

To expand the algebraic expression $(5 x^{2} - 3)^{3}$, we will use the formula $(A - B)^{3} = A^{3} - 3A^{2}B + 3AB^{2} - B^{3}$. This formula is a crucial tool in algebraic expansions, particularly when dealing with cubic expressions. In this instance, we identify $A$ as $5x^2$ and $B$ as $3$. Correctly identifying these components is essential for the accurate application of the expansion formula. The formula provides a systematic way to distribute the cube over the difference of two terms, ensuring that all terms are accounted for. This step-by-step approach not only facilitates the expansion but also enhances comprehension, making it easier to tackle similar problems in the future.

Step 1: Identify A and B

Let $A = 5 x^{2}$ and $B = 3$. This initial step is crucial for setting up the expansion process. By correctly identifying the components of the expression, we ensure that the subsequent calculations are accurate and aligned with the expansion formula. Paying close attention to the terms and their signs is vital for avoiding errors. This meticulous approach forms the basis for a successful expansion.

Step 2: Apply the Formula $(A - B)^{3} = A^{3} - 3A^{2}B + 3AB^{2} - B^{3}$

Substitute $A$ and $B$ into the formula: $(5 x^{2} - 3)^{3} = (5 x^{2})^{3} - 3(5 x^{2})^{2}(3) + 3(5 x^{2})(3)^{2} - (3)^{3}$. This step involves careful substitution of the identified components into the expansion formula. Ensuring that each term is correctly placed within the expansion is essential for maintaining accuracy. The formula provides a systematic way to distribute the cube over the difference of two terms, and its correct application is crucial for obtaining the correct result. This process highlights the importance of algebraic manipulation skills and the ability to follow a structured approach in problem-solving.

Step 3: Calculate $A^{3}$

$(5 x^{2})^{3} = 5^{3} (x^{2})^{3} = 125 x^{6}$. This step involves applying the power of a product rule, which states that $(ab)^n = a^n b^n$, and the power of a power rule, which states that $(a^m)^n = a^{mn}$. By correctly applying these rules, we simplify the cubed term and prepare it for further calculations. Attention to the exponents and coefficients is crucial to avoid errors. This process demonstrates the importance of understanding and applying fundamental exponent rules in algebraic simplification.

Step 4: Calculate $-3A^{2}B$

$-3(5 x^{2})^{2}(3) = -3(25 x^{4})(3) = -225 x^{4}$. In this step, we first square $A$, then multiply the result by -3 and $B$. Accurate multiplication and careful tracking of exponents are essential to ensure the correctness of the result. This process reinforces the importance of algebraic manipulation skills and the ability to combine terms effectively.

Step 5: Calculate $3AB^{2}$

$3(5 x^{2})(3)^{2} = 3(5 x^{2})(9) = 135 x^{2}$. This step involves squaring $B$, then multiplying the result by 3 and $A$. Accurate multiplication and attention to detail are crucial for obtaining the correct result. This step further demonstrates the application of algebraic rules and their role in simplification.

Step 6: Calculate $-B^{3}$

$-(3)^{3} = -27$. This step involves cubing $B$ and applying the negative sign. Accurate calculation is crucial for the final result.

Step 7: Combine All Terms

Putting it all together, we have: $(5 x^{2} - 3)^{3} = 125 x^{6} - 225 x^{4} + 135 x^{2} - 27$. This final step involves combining the results from the previous steps to form the expanded expression. The terms are arranged in a logical order, typically by decreasing powers of the variable. This final expression represents the complete expansion of the original expression and showcases the application of algebraic principles and manipulation skills.

Expanding Expression m: $(x^{m} + x^{n})^{2}$

To expand the algebraic expression $(x^{m} + x^{n})^{2}$, we will once again employ the formula $(A + B)^{2} = A^{2} + 2AB + B^{2}$. This formula is a fundamental tool for expanding binomial squares and is applicable in various algebraic contexts. In this case, we identify $A$ as $x^{m}$ and $B$ as $x^{n}$. By correctly identifying these components, we set the stage for a methodical expansion process that ensures accuracy and clarity. The application of this formula is not merely a mechanical process; it requires a clear understanding of algebraic principles and the ability to recognize patterns within the expression. This step-by-step approach not only facilitates the expansion but also enhances comprehension, making it easier to tackle similar problems in the future.

Step 1: Identify A and B

Let $A = x^{m}$ and $B = x^{n}$. This initial step is crucial for correctly setting up the expansion. Identifying the components of the expression ensures that the subsequent calculations are accurate. It's important to pay close attention to the exponents and variables, as any error in this stage will propagate through the rest of the solution. This meticulous approach helps in avoiding common mistakes and fosters a deeper understanding of the algebraic structure of the expression.

Step 2: Apply the Formula $(A + B)^{2} = A^{2} + 2AB + B^{2}$

Now, we substitute $A$ and $B$ into the formula: $(x^{m} + x^{n})^{2} = (x^{m})^{2} + 2(x^{m})(x^{n}) + (x^{n})^{2}$. This step involves careful substitution and attention to detail to ensure that all terms are correctly placed within the expansion. The expansion formula provides a systematic way to distribute the square over the sum of two terms, and its correct application is essential for obtaining the correct result. This process highlights the importance of algebraic manipulation skills and the ability to follow a structured approach in problem-solving.

Step 3: Calculate $A^{2}$

$(x^{m})^{2} = x^{2m}$. This step involves applying the power of a power rule, which states that $(a^m)^n = a^{mn}$. By correctly applying this rule, we can simplify the squared term and prepare it for further calculations. This process demonstrates the importance of understanding and applying fundamental exponent rules in algebraic simplification.

Step 4: Calculate $2AB$

$2(x^{m})(x^{n}) = 2x^{m+n}$. In this step, we combine like terms by adding their exponents. The product rule for exponents, $a^m \cdot a^n = a^{m+n}$, is applied here to simplify the variable terms. Accurate multiplication and careful tracking of exponents are essential to ensure the correctness of the result. This process reinforces the importance of algebraic manipulation skills and the ability to combine like terms effectively.

Step 5: Calculate $B^{2}$

$(x^{n})^{2} = x^{2n}$. Similar to the calculation of $A^2$, this step involves applying the power of a power rule. Applying the power rule to the variable results in the simplified term. This step further demonstrates the application of exponent rules and their role in algebraic simplification.

Step 6: Combine All Terms

Putting it all together, we have: $(x^{m} + x^{n})^{2} = x^{2m} + 2x^{m+n} + x^{2n}$. This final step involves combining the results from the previous steps to form the expanded expression. The terms are arranged in a logical order, typically by decreasing powers of the variables. This final expression represents the complete expansion of the original expression and showcases the application of algebraic principles and manipulation skills.

Expanding Expression n: $(a^{x} + b^{x+1})^{2}$

To expand the algebraic expression $(a^{x} + b^{x+1})^{2}$, we will utilize the familiar formula $(A + B)^{2} = A^{2} + 2AB + B^{2}$. This formula is a cornerstone of algebraic expansion and is particularly useful when dealing with binomial squares. In this instance, we identify $A$ as $a^{x}$ and $B$ as $b^{x+1}$. Correctly identifying these components is crucial for the accurate application of the expansion formula. The formula provides a systematic way to distribute the square over the sum of two terms, ensuring that all terms are accounted for. This step-by-step approach not only facilitates the expansion but also enhances comprehension, making it easier to tackle similar problems in the future.

Step 1: Identify A and B

Let $A = a^{x}$ and $B = b^{x+1}$. This initial step is crucial for setting up the expansion process correctly. By accurately identifying the components of the expression, we ensure that the subsequent calculations are aligned with the expansion formula. Paying close attention to the terms and their exponents is vital for avoiding errors. This meticulous approach forms the basis for a successful expansion.

Step 2: Apply the Formula $(A + B)^{2} = A^{2} + 2AB + B^{2}$

Substitute $A$ and $B$ into the formula: $(a^{x} + b^{x+1})^{2} = (a^{x})^{2} + 2(a^{x})(b^{x+1}) + (b^{x+1})^{2}$. This step involves careful substitution of the identified components into the expansion formula. Ensuring that each term is correctly placed within the expansion is essential for maintaining accuracy. The formula provides a systematic way to distribute the square over the sum of two terms, and its correct application is crucial for obtaining the correct result. This process highlights the importance of algebraic manipulation skills and the ability to follow a structured approach in problem-solving.

Step 3: Calculate $A^{2}$

$(a^{x})^{2} = a^{2x}$. This step involves applying the power of a power rule, which states that $(a^m)^n = a^{mn}$. By correctly applying this rule, we simplify the squared term and prepare it for further calculations. This process demonstrates the importance of understanding and applying fundamental exponent rules in algebraic simplification.

Step 4: Calculate $2AB$

$2(a^{x})(b^{x+1}) = 2a^{x}b^{x+1}$. In this step, we multiply the terms together. Since the bases are different, we simply write the product as $2a^{x}b^{x+1}$. Accurate multiplication and attention to detail are essential to ensure the correctness of the result. This process reinforces the importance of algebraic manipulation skills and the ability to combine terms effectively.

Step 5: Calculate $B^{2}$

$(b^{x+1})^{2} = b^{2(x+1)} = b^{2x+2}$. Similar to the calculation of $A^2$, this step involves applying the power of a power rule and distributing the exponent. Applying the power rule and distributing the exponent results in the simplified term. This step further demonstrates the application of exponent rules and their role in algebraic simplification.

Step 6: Combine All Terms

Putting it all together, we have: $(a^{x} + b^{x+1})^{2} = a^{2x} + 2a^{x}b^{x+1} + b^{2x+2}$. This final step involves combining the results from the previous steps to form the expanded expression. The terms are arranged in a logical order. This final expression represents the complete expansion of the original expression and showcases the application of algebraic principles and manipulation skills.

Expanding Expression o: $(x^{a+1} + y^{x-2})^{2}$

To expand the algebraic expression $(x^{a+1} + y^{x-2})^{2}$, we will again use the formula $(A + B)^{2} = A^{2} + 2AB + B^{2}$. This formula is a fundamental tool in algebraic expansions and is particularly useful when dealing with binomial squares. In this instance, we identify $A$ as $x^{a+1}$ and $B$ as $y^{x-2}$. Correctly identifying these components is crucial for the accurate application of the expansion formula. The formula provides a systematic way to distribute the square over the sum of two terms, ensuring that all terms are accounted for. This step-by-step approach not only facilitates the expansion but also enhances comprehension, making it easier to tackle similar problems in the future.

Step 1: Identify A and B

Let $A = x^{a+1}$ and $B = y^{x-2}$. This initial step is crucial for correctly setting up the expansion process. By accurately identifying the components of the expression, we ensure that the subsequent calculations are aligned with the expansion formula. Paying close attention to the terms and their exponents is vital for avoiding errors. This meticulous approach forms the basis for a successful expansion.

Step 2: Apply the Formula $(A + B)^{2} = A^{2} + 2AB + B^{2}$

Substitute $A$ and $B$ into the formula: $(x^{a+1} + y^{x-2})^{2} = (x^{a+1})^{2} + 2(x^{a+1})(y^{x-2}) + (y^{x-2})^{2}$. This step involves careful substitution of the identified components into the expansion formula. Ensuring that each term is correctly placed within the expansion is essential for maintaining accuracy. The formula provides a systematic way to distribute the square over the sum of two terms, and its correct application is crucial for obtaining the correct result. This process highlights the importance of algebraic manipulation skills and the ability to follow a structured approach in problem-solving.

Step 3: Calculate $A^{2}$

$(x^{a+1})^{2} = x^{2(a+1)} = x^{2a+2}$. This step involves applying the power of a power rule and distributing the exponent. By correctly applying these rules, we simplify the squared term and prepare it for further calculations. This process demonstrates the importance of understanding and applying fundamental exponent rules in algebraic simplification.

Step 4: Calculate $2AB$

$2(x^{a+1})(y^{x-2}) = 2x^{a+1}y^{x-2}$. In this step, we multiply the terms together. Since the bases are different, we simply write the product as $2x^{a+1}y^{x-2}$. Accurate multiplication and attention to detail are essential to ensure the correctness of the result. This process reinforces the importance of algebraic manipulation skills and the ability to combine terms effectively.

Step 5: Calculate $B^{2}$

$(y^{x-2})^{2} = y^{2(x-2)} = y^{2x-4}$. Similar to the calculation of $A^2$, this step involves applying the power of a power rule and distributing the exponent. Applying the power rule and distributing the exponent results in the simplified term. This step further demonstrates the application of exponent rules and their role in algebraic simplification.

Step 6: Combine All Terms

Putting it all together, we have: $(x^{a+1} + y^{x-2})^{2} = x^{2a+2} + 2x^{a+1}y^{x-2} + y^{2x-4}$. This final step involves combining the results from the previous steps to form the expanded expression. The terms are arranged in a logical order. This final expression represents the complete expansion of the original expression and showcases the application of algebraic principles and manipulation skills.

In conclusion, expanding algebraic expressions requires a strong foundation in algebraic principles and careful application of relevant formulas. Through the detailed examples provided, we have demonstrated step-by-step solutions for expanding expressions involving squares and cubes. Mastery of these techniques is essential for success in algebra and more advanced mathematical studies. Consistent practice and a thorough understanding of algebraic identities will enable you to tackle complex expressions with confidence and accuracy. The ability to expand expressions is not only a valuable skill in itself but also a building block for further mathematical concepts and applications.