Problem 104 - Intermediate Algebra

Question

Perform each indicated operation. See Sections 1.4 and 5.4 $$ \left(9 y^{2}\right)\left(-8 y^{2}\right) $$

Step-by-Step Solution

Answer

The result of the operation is \( -72y^4 \).

Step 1: Understand the operation

Identify the type of mathematical operation we are performing, which in this case is multiplication of two monomials: \( (9y^2) \) and \( (-8y^2) \).

Step 2: Multiply the coefficients

The coefficients 9 and -8 multiply together to give the product: \( 9 \times -8 = -72 \).

Step 3: Multiply the variables

When multiplying variables with the same base, we add their exponents. Here, both terms have the base \( y \) with exponents 2, so you add the exponents: \( y^{2+2} = y^4 \).

Step 4: Combine the results

Combine the results of the coefficient multiplication and variable multiplication to form the final answer: \( -72y^4 \).

Key Concepts

Multiplication of Monomials

When multiplying monomials, you are dealing with terms that consist of a number (known as a coefficient) and a variable raised to an exponent. To multiply two monomials, follow these simple steps:

• First, identify the coefficients and multiply them together.
• Next, look at the variables. If they have the same base, you'll add the exponents together.
• Finally, combine the results to form a new monomial.

For example, in the expression \((9y^2) \times (-8y^2)\), you multiply the coefficients 9 and -8 to get -72. Then, since both terms have the variable \(y\) with exponents 2, you add these exponents: \(y^{2+2} = y^4\). The resulting monomial is \(-72y^4\). This process efficiently simplifies any multiplication involving monomials.

Understanding Exponents

Exponents are a shorthand way to express repeated multiplication of a number or variable by itself. When we talk about exponents, we're basically asking how many times a number (or base) is multiplied by itself. In the expression \(y^2\), the \(2\) is the exponent, and it means \(y\) is multiplied by itself once: \(y \times y\).

One crucial rule when working with exponents, especially in multiplication, is that you add the exponents of like bases. For example:

• \(x^a \times x^b = x^{a+b}\)
• This means if you're multiplying \(y^2\) by \(y^2\), you simply add the 2 and 2 to get \(y^4\).

Remember, this rule only applies to multiplying terms with the same base. Exponents simplify expressions and provide clarity when dealing with large or repetitive multiplications.

Coefficients in Monomials

Coefficients are the numerical parts of monomials. They represent how many times the variable part is counted. In the monomial \(9y^2\), 9 is the coefficient.

• Coefficients can be positive or negative numbers.
• As seen in the expression \((9y^2) \times (-8y^2)\), you multiply the coefficients just like regular numbers.
• In this case, \(9 \times -8 = -72\).

It's important to handle the signs of coefficients carefully. A positive multiplied by a negative result in a negative. The result after multiplying the coefficients gives a new number that quantifies the new variable expression.

Variables in Algebraic Expressions

Variables are symbols used to represent unknown values in expressions and equations, most commonly denoted using letters like \(x, y, z\), etc. In the monomial \(9y^2\), \(y\) is the variable. Variables are crucial in algebra as they allow expressions to generalize mathematical formulas or patterns.

• Variables often come with exponents, detailing the power to which the variable is raised.
• They can be manipulated according to specific algebraic rules.
• In our example, both monomials share the variable \(y\). This makes it easier to simplify by applying exponent rules.

Understanding how variables function and interact with coefficients and exponents is key to mastering algebraic operations, such as the multiplication seen in monomials.