Find The Product: $-8x^5y^2 \cdot 6x^2y$A. $-8x^{10}y^2$ B. $-8x^7y^3$ C. $-48x^7y^3$ D. $-48x^{10}y^2$

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will focus on simplifying a specific type of algebraic expression: the product of two or more variables raised to certain powers. We will use the example of the product −8x5y2⋅6x2y-8x^5y^2 \cdot 6x^2y to demonstrate the step-by-step process of simplifying algebraic expressions.

Understanding the Rules of Exponents

Before we dive into the example, it's essential to understand the rules of exponents. When multiplying variables with the same base, we add the exponents. For example, x2⋅x3=x2+3=x5x^2 \cdot x^3 = x^{2+3} = x^5. When multiplying variables with different bases, we multiply the coefficients and add the exponents. For example, 2x2⋅3y3=6x2y32x^2 \cdot 3y^3 = 6x^2y^3.

Simplifying the Product −8x5y2⋅6x2y-8x^5y^2 \cdot 6x^2y

Now that we have a solid understanding of the rules of exponents, let's simplify the product −8x5y2⋅6x2y-8x^5y^2 \cdot 6x^2y. To do this, we will follow the order of operations (PEMDAS):

  1. Multiply the coefficients: −8⋅6=−48-8 \cdot 6 = -48
  2. Add the exponents of the x terms: 5+2=75 + 2 = 7
  3. Add the exponents of the y terms: 2+1=32 + 1 = 3
  4. Write the simplified expression: −48x7y3-48x^7y^3

Comparing the Simplified Expression to the Answer Choices

Now that we have simplified the expression, let's compare it to the answer choices:

  • A. −8x10y2-8x^{10}y^2
  • B. −8x7y3-8x^7y^3
  • C. −48x7y3-48x^7y^3
  • D. −48x10y2-48x^{10}y^2

The only answer choice that matches our simplified expression is C. −48x7y3-48x^7y^3.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a solid understanding of the rules of exponents. By following the order of operations and applying the rules of exponents, we can simplify complex expressions and arrive at the correct answer. In this article, we used the example of the product −8x5y2⋅6x2y-8x^5y^2 \cdot 6x^2y to demonstrate the step-by-step process of simplifying algebraic expressions.

Tips and Tricks

  • When simplifying algebraic expressions, always follow the order of operations (PEMDAS).
  • When multiplying variables with the same base, add the exponents.
  • When multiplying variables with different bases, multiply the coefficients and add the exponents.
  • Always check your work by comparing the simplified expression to the answer choices.

Practice Problems

  1. Simplify the expression 2x3y2⋅4x2y32x^3y^2 \cdot 4x^2y^3.
  2. Simplify the expression −3x2y4⋅2x3y2-3x^2y^4 \cdot 2x^3y^2.
  3. Simplify the expression 5x4y3⋅3x2y25x^4y^3 \cdot 3x^2y^2.

Answer Key

  1. 8x5y58x^5y^5
  2. −6x5y6-6x^5y^6
  3. 15x6y515x^6y^5

References

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Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when simplifying algebraic expressions. The acronym PEMDAS is commonly used to remember the order of operations:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an algebraic expression with multiple variables?

A: To simplify an algebraic expression with multiple variables, follow these steps:

  1. Identify the variables and their exponents.
  2. Multiply the coefficients (numbers in front of the variables).
  3. Add the exponents of the variables with the same base.
  4. Write the simplified expression.

Q: What is the rule for multiplying variables with the same base?

A: When multiplying variables with the same base, add the exponents. For example, x2⋅x3=x2+3=x5x^2 \cdot x^3 = x^{2+3} = x^5.

Q: What is the rule for multiplying variables with different bases?

A: When multiplying variables with different bases, multiply the coefficients and add the exponents. For example, 2x2⋅3y3=6x2y32x^2 \cdot 3y^3 = 6x^2y^3.

Q: How do I simplify an expression with negative exponents?

A: To simplify an expression with negative exponents, follow these steps:

  1. Rewrite the expression with positive exponents by moving the variable to the other side of the fraction.
  2. Simplify the expression.

Q: What is the rule for dividing variables with the same base?

A: When dividing variables with the same base, subtract the exponents. For example, x3÷x2=x3−2=x1x^3 \div x^2 = x^{3-2} = x^1.

Q: What is the rule for dividing variables with different bases?

A: When dividing variables with different bases, divide the coefficients and subtract the exponents. For example, 2x3÷3y2=23x3y−22x^3 \div 3y^2 = \frac{2}{3}x^3y^{-2}.

Q: How do I simplify an expression with fractions?

A: To simplify an expression with fractions, follow these steps:

  1. Simplify the numerator and denominator separately.
  2. Multiply the numerator and denominator by the reciprocal of the denominator.
  3. Simplify the resulting expression.

Q: What is the rule for simplifying expressions with absolute values?

A: When simplifying expressions with absolute values, follow these steps:

  1. Evaluate the expression inside the absolute value.
  2. If the result is positive, write it as is.
  3. If the result is negative, write it as a negative number.

Q: How do I simplify an expression with radicals?

A: To simplify an expression with radicals, follow these steps:

  1. Simplify the expression inside the radical.
  2. If the result is a perfect square, simplify the radical.
  3. If the result is not a perfect square, leave the radical as is.

Q: What is the rule for simplifying expressions with exponents and radicals?

A: When simplifying expressions with exponents and radicals, follow these steps:

  1. Simplify the expression inside the radical.
  2. Simplify the expression with exponents.
  3. Combine the simplified expressions.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a solid understanding of the rules of exponents, variables, and operations. By following the order of operations and applying the rules of exponents, we can simplify complex expressions and arrive at the correct answer. In this article, we have covered frequently asked questions and provided step-by-step solutions to common problems.