Problem 78 - Algebra A Combined Function

Question

Multiply. $$ -z^{2} y(11 z y) $$

Step-by-Step Solution

Answer

\(-11z^3y^2\)

Step 1: Identify the Terms to be Multiplied

The expression given is \(-z^{2} y(11 z y)\).Here, you can identify two terms to be multiplied:1. \(-z^2 y\)2. \(11 z y\).These are the terms that we will multiply together.

Step 2: Expand the Multiplication

When multiplying terms, multiply the coefficients, the powers of \(z\), and the powers of \(y\) separately.The expression is:\((-1) \, \cdot \, 11 \)for the coefficients, then:\(z^2 \, \cdot \, z \)for the \(z\) terms, and:\(y \, \cdot \, y\)for the \(y\) terms.

Step 3: Multiply the Coefficients

Coefficients are numbers without variables. The coefficients are \(-1\)and \(11\).Multiply these together:\(-1 \, \cdot \, 11 = -11\).

Step 4: Multiply the Powers of z

The powers of \(z\) are \(z^2\)and \(z^1\).When multiplying, add the exponents:\(z^{2+1} = z^3\).

Step 5: Multiply the Powers of y

The powers of \(y\) are \(y\)and \(y^1\).When multiplying, also add the exponents:\(y^{1+1} = y^2\).

Step 6: Combine All Results

Combine the results from Steps 3, 4, and 5:Coefficient: \(-11\)\(z\) term: \(z^3\)\(y\) term: \(y^2\)Thus, the final expression is:\(-11z^3y^2\).

Key Concepts

Coefficients

In algebraic expressions, coefficients are the numeric parts that stand next to variables. They give the variable a specific magnitude or size. For instance, in the term \(-z^2y\), the coefficient is \-1\. Similarly, in the term \(11zy\), the coefficient is \11\.

When multiplying coefficients from different terms, you simply multiply the numbers together. If we take \-1\ and \11\ from our example, the multiplication would be \(-1 \cdot 11 = -11\).

Coefficients are crucial because they affect the overall size of the resulting values in the expression. Negative signs are important here as they change the result altogether.

Powers of Variables

When dealing with multiple terms, it is important to handle the powers of variables properly. A power indicates how many times a number or a variable is multiplied by itself. For example, in the term \(z^2\), the \2\ represents the power and means \(z\) is multiplied by itself twice, \(z \cdot z\).

Each variable can have its power, for example, in \(11zy\), \(z^1\) is implied. This shows that \(z\) stands alone. When multiplying variables with powers, you add the powers together. For instance, combining \(z^2\) and \(z\) (which is \(z^1\)), you get \(z^{2+1} = z^3\).

This understanding is key to simplifying expressions and performing operations correctly.

Exponents

Exponents are the numbers that tell you how many times to multiply a base number by itself. When dealing with algebraic expressions, exponents play a vital role, especially in the multiplication of terms. Consider the expression \(z^2\), where \(2\) is the exponent indicating that \(z\) is used twice in multiplication (i.e., \(z \cdot z\)).

When multiplying terms with the same base, such as \(-z^2y(11zy)\), you add the exponents of each variable. This is done by summing up the exponents of the variables \(z\) and \(y\).- For \(z\): \(z^2 \, \cdot \, z^1 = z^{2+1} = z^3\)- For \(y\): \(y^1 \, \cdot \, y^1 = y^{1+1} = y^2\)

This process enables simplification and clarity in expressions.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can involve numbers, variables, and operations. For instance, \(-z^2y(11zy)\) is an algebraic expression consisting of multiplication among terms with both coefficients and variables.

These expressions can be combined and simplified through rules and properties, like when multiplying terms or adding exponents. Understanding each component allows you to manipulate the expression effectively. This includes recognizing:- The coefficients, which affect the scale of the expression- The variables and their associated powers, which dictate how terms are structured- The operations being used, such as multiplication or addition, and their respective rules

Practicing with algebraic expressions sharpens problem-solving skills and fosters a deeper comprehension of mathematical concepts.