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Chapter 5: Problem 96

Multiply. See Section 5.1 $$ -z^{2} y(11 z y) $$

Short Answer

Expert verified
-11z^3 y^2.

Step by step solution

01

Distribute the Terms

To multiply the expression \[-z^{2} y(11 z y),\]we need to distribute the term \(-z^2 y\) to \(11 z y\). This involves multiplying each part of \(-z^2 y\) by each part of \(11 z y\).
02

Multiply the Coefficients

Start by multiplying the coefficients (numbers) from each term. The coefficients are -1 (from -\(z^2 y\)) and 11 from (11 \(z y\)). Multiply these to get:\(-1 \times 11 = -11.\)
03

Multiply the Variables

Next, multiply the variables. For \(-z^2 y\) and \(11 z y\), we multiply:\(z^2 \times z = z^{2+1} = z^3\),and \(y \times y = y^{1+1} = y^2.\)
04

Combine the Results

Combine the results from Steps 2 and 3 to form the final expression. We have:\(-11\), the coefficient from Step 2; \(z^3\), the variable result from Step 3; and \(y^2\), the other variable result from Step 3.Thus, the final expression is: \(-11 z^3 y^2.\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
When facing an expression like \(-z^{2} y(11 z y)\), understanding the distributive property is crucial. The distributive property allows you to multiply a single term by each term inside a parenthesis. Consider the expression \(-z^2 y(11 z y)\). Here, you are asked to distribute or "spread out" the multiplication, essentially multiplying \(-z^2 y\)by \(11 z y\) term by term.
  • First, take \(\-z^2 y\), which acts as the distributor.
  • Multiply it with the first component of \(11 z y\), which is \(11 z\).
  • Next, multiply \(-z^2 y\) by the second component, \( y\).
This approach ensures every part of the parentheses has been touched by the distributor, leading to a complete multiplication of terms.
Multiplying Coefficients
When you encounter coefficients, like the numeric parts in algebraic terms, you'll find it's generally straightforward. In this exercise, the coefficients are \(-1\) (from \(-z^2 y\)) and \(11\)(from \(11 z y\)).
These numbers help quantify the variables. To multiply the coefficients, simply multiply the numbers:
  • The calculation is \(-1 \times 11 = -11\).
At this stage, you're only focusing on the numeric part; it's simply basic multiplication of numbers. This step is key because it forms the numerical base of your final expression. Remembering to handle any negative signs properly is crucial in achieving the correct result.
Exponent Rules
Handling exponents correctly is central to simplifying multiplication of variables. Exponents tell you how many times a number or a variable is used in multiplication. In the problem \(-z^{2} y(11 z y)\), let's look at the exponents:
  • When multiplying \(z^2\) and \(z\),use the rule \(z^{a} \times z^{b} = z^{a+b}\). It gives \(z^{2+1} = z^3\).
  • Similarly, for \(y \times y\), the result is \(y^{1+1} = y^2\).
These simple rules make multiplying variables manageable by consolidating the operations into a single, simplified exponent. Adding the exponents during multiplication is a critical skill and ensures you derive the right expression efficiently. It's about keeping track of how many times each variable appears in the product.

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Most popular questions from this chapter

Add or subtract as indicated. See Examples 9 through \(13 .\) $$ \left(4 x^{2}+y^{2}+3\right)-\left(x^{2}+y^{2}-2\right) $$

Multiply. See Example 7. $$ (4 x+5)(4 x-5) $$

Multiply. $$(4 x+5)\left(8 x^{2}+2 x-4\right)$$

Fill in the boxes so that each statement is true. (More than one answer is possible for these exercises). $$ x^{\square}=\frac{1}{x^{5}} $$

Square where indicated. Simplify if possible. $$(5 x+2 y)^{2}$$
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