Question

Multiply and simplify. $$(3.73 x y)\left(1.77 x y^{2}\right)$$

Short answer

The simplified product is \(6.6021x^2y^3\).

Step-by-step solution

Multiply the Coefficients

First, multiply the numerical coefficients (the numbers) from each term together. In this case, multiply 3.73 by 1.77.

Multiply the Literal Coefficients

Next, multiply the literal coefficients (the variables) from each term together. Multiply the x's together and the y's together, remembering that when you multiply variables with the same base, you add their exponents.

Combine the Products

Combine the products of the numerical coefficients and the literal coefficients to get the simplified expression.

Key concepts

Multiplying Coefficients

When simplifying algebraic expressions, multiplying coefficients is a key step you'll encounter. Coefficients are the numerical parts of terms in an algebraic expression. To multiply coefficients, you simply multiply the numbers as you would in basic arithmetic. For example, if you're given the expression \(3.73 \times y\times 1.77 \times y^2\), you start by multiplying the numbers 3.73 and 1.77 together.

This operation will give you a new coefficient for the resulting term. Ensuring that you accurately multiply these numbers is crucial as it affects the entire outcome of the expression.

Exponent Rules

Understanding exponent rules is fundamental to simplifying expressions with variables raised to a power, like \(x^m \times x^n\). When you multiply variables that have the same base, you add their exponents—the powers to which the variable is raised. This is known as the Product of Powers rule.

Applying the Exponent Rule

For the given problem, \(x \times x\) becomes \(x^{1+1}\) or \(x^2\), and similarly, \(y \times y^2\) becomes \(y^{1+2}\) or \(y^3\). It's also essential to know that a variable without an explicit exponent actually has an exponent of one. Remember, these rules are not just limited to variables but apply to any base that is raised to a power.

Combining Like Terms

Once we've dealt with coefficients and exponents, we often need to combine like terms to simplify the expression further. Like terms are terms that have the exact same variables raised to the same powers. While our current example does not have like terms that need to be combined after multiplication, it's useful to note that this is a common final step. If you're simplifying an expression and find multiple terms with the same variable parts, you will add or subtract the coefficients of these terms to combine them.

For instance, if you end up with terms like \(5x^2y\) and \(3x^2y\) in an expression, you can add the coefficients (5 and 3) to get \(8x^2y\), since \(x^2y\) is the common term. It's like saying, 'I have 5 apples and 3 apples, so I have 8 apples total.' Combining like terms streamlines the expression, making it easier to work with or evaluate.