Problem 110 Simplify by combining like terms... [FREE SOLUTION]

91影视

Understanding Elementary Algebra with Geometry

Lewis Hirsch, Arthur Goodman

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 2: Problem 110

Simplify by combining like terms whenever possible. $$7 y^{2}(y+2)+2 y\left(5 y^{2}\right)$$

Short Answer

Expert verified

The simplified expression is $17y^3 + 14y^2$.

Step by step solution

Distribute the terms inside the parentheses

First, simplify the expression inside each set of parentheses: For the term $7y^2(y+2)$, distribute $7y^2$ to both $y$ and $2$: $7y^2 \times y + 7y^2 \times 2 = 7y^3 + 14y^2$.

Simplify second term inside parentheses

Now, simplify the second term $2y(5y^2)$ by distributing $2y$ to $5y^2$: $2y \times 5y^2 = 10y^3$.

Combine the simplified terms

Combine the simplified terms from Step 1 and Step 2 to form the new expression: $7y^3 + 14y^2 + 10y^3$.

Combine like terms

Identify and combine like terms: Combine the $y^3$ terms: $7y^3 + 10y^3 = 17y^3$.This gives us: $17y^3 + 14y^2$.

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Key Concepts

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

Combining Like Terms

Combining like terms is an essential step in simplifying algebraic expressions. It involves merging terms that have identical variable components. For example, in the expression $ 7y^3 + 14y^2 + 10y^3 $, the terms $ 7y^3 $ and $ 10y^3 $ are like terms because they both contain $ y^3 $.
Here are the steps to combine like terms:

Identify terms with the same variables raised to the same powers.
Add or subtract the coefficients of these terms while keeping the variable part unchanged.

For instance, combining $ 7y^3 $ and $ 10y^3 $ gives $ 7 + 10 = 17 $, resulting in $ 17y^3 $. The result from our combination step is: $ 17y^3 + 14y^2 $.
This is the simplest form of the given expression, achieved by combining like terms.

Distributive Property

The distributive property is a powerful tool in algebra that helps to simplify expressions and solve equations. This property states that for any numbers or variables $ a $, $ b $, and $ c $, the equation $ a(b + c) = ab + ac $ holds true.
In our exercise, we apply the distributive property as follows:

For $ 7y^2(y + 2) $, distribute $ 7y^2 $ to both $ y $ and $ 2 $: $ 7y^2 \times y + 7y^2 \times 2 = 7y^3 + 14y^2 $.
For $ 2y(5y^2) $, distribute $ 2y $ to $ 5y^2 $: $ 2y \times 5y^2 = 10y^3 $.

Using the distributive property simplifies our task of breaking down complex expressions, making it easier to then combine like terms.

Exponentiation

Exponentiation involves raising numbers or variables to the power of another number, called the exponent. This operation is written as $ x^n $, where $ x $ is the base and $ n $ is the exponent. Understanding exponentiation is crucial for dealing with algebraic expressions.
In our exercise, terms like $ 7y^3 $, $ 14y^2 $, and $ 10y^3 $ all involve exponentiation:

$ y^3 $ means $ y \times y \times y $.
$ y^2 $ means $ y \times y $.

During the process of distribution, the exponents come into play when multiplying variables:

For instance, $ 7y^2 \times y = 7y^{2+1} = 7y^3 $.
Similarly, $ 2y \times 5y^2 = 10y^{1+2} = 10y^3 $.

The exponents add when multiplying like bases, reinforcing the importance of properly handling exponentiation in algebra.

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91影视

Simplify by combining like terms whenever possible. $$7 y^{2}(y+2)+2 y\left(5 y^{2}\right)$$

Short Answer

Step by step solution

Distribute the terms inside the parentheses

Simplify second term inside parentheses

Combine the simplified terms

Combine like terms

Key Concepts

Combining Like Terms

Distributive Property

Exponentiation

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