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Chapter 5: Problem 14

Use the product rule to simplify each expression. See Examples I and 2. $$ (-7 x y)(7 y) $$

Short Answer

Expert verified
The simplified expression is \(-49xy^2\).

Step by step solution

01

Identify the Terms

The expression given is \((-7xy)(7y)\). We need to identify the two separate terms to apply the product rule for multiplication. Here, \(-7xy\) is the first term and \(7y\) is the second term.
02

Multiply the Coefficients

Multiply the numerical coefficients of each term together. The coefficients we have are \(-7\) and \(7\). Calculate \(-7 \times 7 = -49\).
03

Multiply the Variables

Combine the variables from both terms \(x\), \(y\), and \(y\). Multiply the common base \(y\) by adding their exponents together: \(y^1 \times y^1 = y^{1+1} = y^2\). Therefore, the variables multiplied together are \(xy^2\).
04

Combine the Results

Now, combine the results from Steps 2 and 3: \(-49xy^2\). This is the simplified expression after using the product rule.

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is a critical skill in algebra that involves making mathematical expressions more concise and manageable. In the expression \((-7xy)(7y)\), simplifying means to combine like terms and reduce the expression to its simplest form.
By applying mathematical rules, such as the product rule, you can transform complex expressions into simpler ones.
  • First, identify the components of the expression.
  • Use applicable rules like distributing or combining,
  • Finally, form a genuine understanding of the expression.
Simplifying doesn't change the value but presents it in a less complicated way, making further calculations easier.
Multiplying Coefficients
The coefficients in a term are the numerical values multiplied by the variables. In the expression \((-7xy)(7y)\), they are \(-7\) and \(7\).
To simplify, full focus first goes on multiplying these coefficients. Here鈥檚 how it works:

  • Multiply the coefficients \(-7\) and \(7\), leading to \(-49\).
Multiplying coefficients is straightforward: you simply multiply the numbers as you normally would, regardless of their signs. A negative times a positive equals a negative, so \(-49\) becomes our new coefficient for the simplified expression.
Combining Variables
When combining variables, you focus on grouping and simplifying them based on their bases and exponents.
Taking each variable in the expression \((-7xy)(7y)\):
  • Identify the variable base. Here, we have base \(x\) and base \(y\).
  • Combine like bases together.
For the expression \((-7xy)(7y)\), we see \(x\) stays as is since \(7y\) doesn't have it.
The \(y\) in both terms combine using exponent rules, discussed in the next section.
Exponent Addition
Exponent addition is a powerful rule when combining terms with the same base. In this exercise, you must correctly apply the rule to the expression \((-7xy)(7y)\):
  • The \(y\) variable appears in both terms: \(y^1\) and \(y^1\).
  • Add the exponents: \(1 + 1 = 2\).
Thus, it results in \(y^2\).
The rule is simple: when multiplying terms with the same base, add the exponents.
This helps you maintain accurate powers in simplified expressions without changing their inherent values.

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Most popular questions from this chapter

If \(P(x)\) is the polynomial given, find a. \(P(a),\) b. \(P(-x),\) and c. \(P(x+h)\). \(P(x)=3 x-2\)

Recall that a graphing calculator may be used to check addition, subtraction, and multiplication of polynomials. In the same manner, a graphing calculator may be used to check factoring of polynomials in one variable. For example, to see that $$ 2 x^{3}-9 x^{2}-5 x=x(2 x+1)(x-5) $$ graph \(\mathrm{Y}_{1}=2 x^{3}-9 x^{2}-5 x\) and \(\mathrm{Y}_{2}=x(2 x+1)(x-5) .\) Then trace along both graphs to see that they coincide. Factor the following and use this method to check your results. $$ 30 x^{3}+9 x^{2}-3 x $$

Factor each polynomial completely. See Examples 1 through 12. $$ 2(x+4)^{2}+3(x+4)-5 $$

If \(P(x)\) is the polynomial given, find a. \(P(a),\) b. \(P(-x),\) and c. \(P(x+h)\). \(P(x)=8 x+3\)

Factor each polynomial completely. See Examples 1 through 12. $$ 4 x^{2}+12 x+9 $$
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