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Chapter 5: Problem 86

Simplify each expression. See Examples 1 through 11. $$ \left(3 y^{4}\right)(-5 y) $$

Short Answer

Expert verified
The simplified expression is \(-15y^5\).

Step by step solution

01

Distribute the Coefficients

First, multiply the coefficients (the numbers in front of the variables). We have 3 times -5, which gives us \[ 3 imes (-5) = -15. \]
02

Apply the Product of Powers Rule

When multiplying like bases, we add the exponents. Here, the base is \( y \). Write the product of the powers as follows: \[ y^4 imes y^1 = y^{4+1} = y^5. \]
03

Combine Results

Combine the results from the previous steps to write the simplified expression. Multiply the results: \[ -15 \times y^5. \] The simplified expression is \[ -15y^5. \]

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Key Concepts

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

Coefficients
Coefficients are the numbers that multiply the variable parts in algebraic expressions. They give us crucial information about the scale or quantity of the variables involved. In our example, we have the coefficients 3 and -5 in the expression \((3y^4)(-5y)\). These numbers tell us how many times we are considering the accompanying variable terms. To simplify an expression involving coefficients, you begin by performing arithmetic operations on these numbers. For multiplication, simply multiply them together.
  • In this case, multiply 3 and -5 to get \(-15\).
Working with coefficients is essential for simplifying algebraic expressions, as they impact the overall result of the operation performed on the variables.
Product of Powers Rule
The product of powers rule is an important tool in algebra for simplifying expressions involving exponents. This rule applies when multiplying terms that have the same base. Instead of multiplying each individually, you simply add the exponents together. For example, in our exercise the terms \(y^4\) and \(y\) share the base \(y\). Instead of writing it out in its entirety, you just add the exponents:
  • For \(y^4 \times y^1\), you add the exponents 4 and 1 to get \(y^{4+1} = y^5\).
This rule greatly simplifies multiplication of variables and makes calculations easier and faster. By understanding and applying the product of powers rule, you gain efficiency in solving polynomial expressions.
Distributive Property
The distributive property is a handy algebraic rule used to multiply a single term across terms inside a parenthesis. It tells us to multiply each term inside the parenthesis by the term outside.Even though our exercise focuses on coefficients and powers, the initial "distribution" is considered when we separate and multiply coefficients and like bases. Here's what happens:
  • Given the expression \((3y^4)(-5y)\), apply a sort of distributive logic: First, multiply the coefficients \((3)(-5)\) and then multiply the terms \(y^4\) and \(y\).
This approach keeps operations organized and simplifies the expression step by step. As you become more adept at using this property, you'll find it eases complexity, whether in simplifying expressions or solving equations in algebra.

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Most popular questions from this chapter

The volume of the swimming pool shown is \(\left(36 x^{5}-12 x^{3}\right.\) \(\left.+6 x^{2}\right)\) cubic feet. If its height is \(2 x\) feet and its width is \(3 x\) feet, find its length.

Write each number in standard notation. See Example 6 Each second, the Sun converts \(7.0 \times 10^{8}\) tons of hydrogen into helium and energy in the form of gamma rays. Write this number in standard notation. (Source: Students for the Exploration and Development of Space)

Add or subtract as indicated. See Examples 9 through \(13 .\) $$ \left(x^{2}+2 x y-y^{2}\right)+\left(5 x^{2}-4 x y+20 y^{2}\right) $$

Match each expression on the left to the equivalent expression on the right. See the Concept Check in this section. $$ (a-b)^{2} $$ $$A. a^{2}-b^{2}$$ $$B. a^{2}+b^{2}$$ $$C. a^{2}-2 a b+b^{2}$$ $$D. a^{2}+2 a b+b^{2}$$ $$E. none of these$$

Multiply each expression. $$ 2 a\left(a^{2}+1\right) $$
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