Problem 71 $(a+b)(a-b)=a^{2}-b^{2}$... [FREE SOLUTION]

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Elementary Algebra

Laura Bracken, Ed Miller

$Math Studyset 91影视 Explanations$ Math

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Chapter 5: Problem 71

$(a+b)(a-b)=a^{2}-b^{2}$

Short Answer

Expert verified

(a+b)(a-b) = a^2 - b^2

Step by step solution

Distribute the Terms

Apply the distributive property to the left-hand side of the equation: $(a + b)(a - b) = a \times a + a \times (-b) + b \times a + b \times (-b)$.

Simplify the Products

Simplify the products from the previous step: $a \times a = a^2$, $a \times (-b) = -ab$, $b \times a = ba$, and $b \times (-b) = -b^2$. This results in: $a^2 - ab + ba - b^2$.

Combine Like Terms

Notice that $-ab + ba = 0$ (since $ba = ab$). This simplifies the expression to $a^2 - b^2$.

Verification

Verify that the simplified expression $a^2 - b^2$ matches the right-hand side of the given equation $(a + b)(a - b) = a^2 - b^2$.

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Key Concepts

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

Distributive property

The distributive property is a fundamental concept in algebra. It allows you to simplify expressions by distributing multiplication over addition or subtraction. For example, in the expression $a(b + c)$, you can apply the distributive property to get $ab + ac$.

In our exercise, we applied the distributive property to $a + b$ and $a - b$. This means multiplying each term in the first parentheses with each term in the second parentheses. So, $a + b$ times $a - b$ turned into:
\[ (a + b)(a - b) = a \times a + a \times (-b) + b \times a + b \times (-b) \]
Each term in the first group is multiplied by each term in the second group.

Simplifying expressions

After distributing the terms in our algebraic expression, we need to simplify the products. Simplification makes the expression more manageable and easier to work with. In this exercise, simplifying means calculating each of the four products obtained from the distributive property.

We simplify as follows:

$a \times a = a^2$
$a \times (-b) = -ab$
$b \times a = ba$, which is the same as $ab$
$b \times (-b) = -b^2$

After simplification, we get the expression: $a^2 - ab + ba - b^2$.

Combining like terms

The next step in simplifying an expression is to combine like terms. Terms are like terms if they have the same variables raised to the same power.

In the expression $a^2 - ab + ba - b^2$, we identify $ab$ and $ba$ as like terms. Remember that $ba$ is the same as $ab$.

Combining these, we get:
\[ -ab + ab = 0 \]
So, our expression simplifies further to:
\[ a^2 - b^2 \] This matches the right-hand side of our original equation: \[ (a + b)(a - b) = a^2 - b^2 \] Combining like terms ensures the expression is in its simplest form.

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91影视

\((a+b)(a-b)=a^{2}-b^{2}\)

Short Answer

Step by step solution

Distribute the Terms

Simplify the Products

Combine Like Terms

Verification

Key Concepts

Distributive property

Simplifying expressions

Combining like terms

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