What Is The Product Of (9)(-9)(-1) A Comprehensive Guide

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In the realm of mathematics, particularly in basic arithmetic and algebra, understanding how to multiply integers is a fundamental skill. This article delves into the process of multiplying integers, focusing on a specific problem: finding the product of $(9)(-9)(-1)$. This seemingly simple problem provides an excellent opportunity to review the rules of multiplication involving positive and negative numbers. By breaking down the steps and explaining the underlying principles, we aim to provide a comprehensive understanding of how to solve such problems and similar mathematical challenges.

Breaking Down the Problem: Multiplying Integers

To effectively tackle the problem of finding the product of $(9)(-9)(-1)$, it's essential to first understand the rules governing the multiplication of integers. Integers are whole numbers, which can be positive, negative, or zero. The multiplication of integers follows a specific set of rules related to the signs of the numbers being multiplied.

Rules of Integer Multiplication

The cornerstone of integer multiplication lies in the following rules:

  1. A positive number multiplied by a positive number yields a positive result.
  2. A negative number multiplied by a negative number also yields a positive result.
  3. A positive number multiplied by a negative number (or vice versa) yields a negative result.

These rules are crucial because they dictate the sign of the product. When dealing with multiple numbers, as in our problem, we apply these rules sequentially. Let's see how these rules play out when we start solving $(9)(-9)(-1)$.

Step-by-Step Solution of $(9)(-9)(-1)$

Our problem involves the product of three integers: 99, −9-9, and −1-1. To solve it, we can proceed step by step, multiplying two numbers at a time.

First, let's multiply the first two numbers: 99 and −9-9. According to the rules, a positive number multiplied by a negative number results in a negative number. Therefore,

(9)×(−9)=−81(9) \times (-9) = -81

Now, we have reduced the problem to finding the product of −81-81 and −1-1. We multiply these two numbers:

(−81)×(−1)(-81) \times (-1)

Here, we are multiplying a negative number by another negative number. According to the rules, this yields a positive result. So,

(−81)×(−1)=81(-81) \times (-1) = 81

Thus, the product of $(9)(-9)(-1)$ is 8181. This step-by-step approach not only gives us the answer but also reinforces our understanding of the rules of integer multiplication.

Applying the Rules to Complex Problems

The method we used to solve $(9)(-9)(-1)$ can be applied to more complex problems involving multiple integers. The key is to proceed sequentially, applying the rules of integer multiplication at each step. For example, if we had a problem like $(-2)(3)(-4)(5)$, we would first multiply −2-2 and 33, then multiply the result by −4-4, and finally multiply that result by 55. This process ensures that we correctly account for the signs of the numbers and arrive at the correct product.

Why This Matters: Real-World Applications

Understanding how to multiply integers isn't just a theoretical exercise; it has numerous real-world applications. For example, in finance, you might use integer multiplication to calculate losses and gains. If a stock loses $5 per share, and you own 10 shares, you can use multiplication to find the total loss: $(10) \times (-5) = -50$. Similarly, in physics, you might use integer multiplication to calculate changes in position or velocity. The ability to correctly multiply integers is therefore a valuable skill in many different contexts.

Common Mistakes to Avoid

When multiplying integers, there are a few common mistakes to watch out for. One of the most frequent errors is misapplying the rules of signs. For example, some students may incorrectly assume that a negative times a negative always results in a negative. Another common mistake is forgetting to account for the signs at all, simply multiplying the numbers as if they were all positive. To avoid these errors, it's helpful to write out each step and carefully apply the rules of integer multiplication. Regular practice and careful attention to detail are the best ways to prevent these mistakes.

The Role of Zero in Multiplication

While we've focused on multiplying positive and negative integers, it's important to also understand the role of zero in multiplication. Any number multiplied by zero is zero. This rule applies regardless of whether the number is positive, negative, or zero itself. For example,

5×0=05 \times 0 = 0

(−5)×0=0(-5) \times 0 = 0

0×0=00 \times 0 = 0

This property of zero is fundamental in mathematics and is essential to remember when solving problems involving multiplication.

Conclusion: Mastering Integer Multiplication

In conclusion, mastering the multiplication of integers is a critical skill in mathematics. The problem of finding the product of $(9)(-9)(-1)$ serves as a valuable example for understanding the rules governing the signs of integers in multiplication. By breaking down the problem step by step and applying these rules carefully, we can arrive at the correct answer, which in this case is 8181. Furthermore, the principles discussed here extend to more complex problems and have numerous real-world applications. By understanding these concepts, students can build a solid foundation for further mathematical studies. Regular practice, careful attention to detail, and a thorough understanding of the rules are the keys to success in multiplying integers.

Distributive Property: Expanding Expressions

Another important concept related to multiplication is the distributive property. This property allows us to multiply a single term by multiple terms inside parentheses. It's a fundamental concept in algebra and is essential for simplifying expressions and solving equations. The distributive property states that for any numbers aa, bb, and cc:

a×(b+c)=a×b+a×ca \times (b + c) = a \times b + a \times c

This means that we can distribute the term outside the parentheses to each term inside the parentheses. Let's look at an example:

3×(x+2)3 \times (x + 2)

Using the distributive property, we multiply 33 by each term inside the parentheses:

3×x+3×2=3x+63 \times x + 3 \times 2 = 3x + 6

The distributive property also applies when there is subtraction inside the parentheses:

a×(b−c)=a×b−a×ca \times (b - c) = a \times b - a \times c

For example:

5×(y−4)5 \times (y - 4)

Distributing the 55 gives us:

5×y−5×4=5y−205 \times y - 5 \times 4 = 5y - 20

Applying the Distributive Property with Integers

The distributive property becomes particularly useful when dealing with negative integers. Let's consider an example that combines the distributive property with the rules of integer multiplication:

−2×(3−x)-2 \times (3 - x)

Here, we need to distribute −2-2 to both 33 and −x-x. First, we multiply −2-2 by 33:

(−2)×3=−6(-2) \times 3 = -6

Next, we multiply −2-2 by −x-x. Remember that a negative multiplied by a negative gives a positive:

(−2)×(−x)=2x(-2) \times (-x) = 2x

So, the expression becomes:

−6+2x-6 + 2x

This example illustrates how to apply the distributive property while also keeping the rules of integer multiplication in mind. This skill is crucial for simplifying algebraic expressions and solving equations.

Advanced Applications of the Distributive Property

In more advanced mathematics, the distributive property is used in a variety of contexts, including polynomial multiplication. When multiplying two binomials (expressions with two terms), such as $(x + 2)(x - 3)$, we use a form of the distributive property often referred to as the FOIL method (First, Outer, Inner, Last).

Let's break down how this works:

  1. First: Multiply the first terms in each binomial: $x \times x = x^2$
  2. Outer: Multiply the outer terms: $x \times (-3) = -3x$
  3. Inner: Multiply the inner terms: $2 \times x = 2x$
  4. Last: Multiply the last terms: $2 \times (-3) = -6$

Now, we add all these results together:

x2−3x+2x−6x^2 - 3x + 2x - 6

Finally, we combine like terms (in this case, the terms with xx):

x2−x−6x^2 - x - 6

This process demonstrates how the distributive property is a powerful tool for expanding and simplifying complex expressions. Understanding this property is essential for success in algebra and beyond.

Practice Problems

To solidify your understanding of the distributive property and integer multiplication, let's work through a few practice problems.

  1. Simplify: $4 \times (2y + 5)$
  2. Simplify: $-3 \times (a - 4)$
  3. Expand: $(x + 1)(x + 2)$

Solving these problems will help you practice applying the distributive property in different contexts and reinforce your understanding of the rules of integer multiplication. Remember to break down each problem step by step and pay close attention to the signs of the numbers.

Conclusion: The Distributive Property as a Key Skill

The distributive property is a fundamental concept in mathematics that allows us to simplify expressions and solve equations. By multiplying a single term by multiple terms inside parentheses, we can expand expressions and make them easier to work with. When combined with the rules of integer multiplication, the distributive property becomes an even more powerful tool. From basic algebra to advanced calculus, this property is used extensively in mathematical problem-solving. Mastering the distributive property is therefore essential for any student looking to succeed in mathematics.

Conclusion: Mastering Integer Multiplication and Beyond

In summary, understanding and applying the rules of integer multiplication is a crucial skill in mathematics. The initial problem of finding the product of $(9)(-9)(-1)$ provides a clear example of how these rules work in practice, resulting in the answer 8181. We've also explored the distributive property, another key concept that allows us to simplify expressions and solve equations. By mastering these concepts, students can build a solid foundation for further mathematical studies and tackle more complex problems with confidence. Whether you're dealing with basic arithmetic or advanced algebra, the principles discussed in this article will serve as valuable tools in your mathematical journey. Consistent practice, a clear understanding of the rules, and careful attention to detail are the keys to success in mathematics.