Problem 295
Question
In the following exercises, simplify. $$ (7+\sqrt{10})(7-\sqrt{10}) $$
Step-by-Step Solution
Verified Answer
39.
1Step 1: Identify the Expression
The given expression is .
2Step 2: Recognize the Formula
This is in the form of .
3Step 3: Apply the Difference of Squares Formula
The formula for the difference of squares is . Here, .
4Step 4: Simplify
By applying the difference of squares formula, find that .
5Step 5: Final Step
Simplify the result to arrive at the final answer, .
Key Concepts
Understanding Algebraic ExpressionsThe Difference of SquaresSimplification Steps Made Easy
Understanding Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operators (such as +, -, *, and /). They are crucial for solving various mathematical problems.
An algebraic expression might look something like this: \(3x + 7\) or \(4y^2 - 5y + 6\).
In our exercise, we have the expression \((7+\sqrt{10})(7-\sqrt{10})\).
This is a bit more complex, but understanding the basics helps us identify the necessary steps for simplification.
When simplifying algebraic expressions, the goal is to find an equivalent expression that is easier to work with. This can involve combining like terms, factoring, or using algebraic identities (which is the focus of this problem).
An algebraic expression might look something like this: \(3x + 7\) or \(4y^2 - 5y + 6\).
In our exercise, we have the expression \((7+\sqrt{10})(7-\sqrt{10})\).
This is a bit more complex, but understanding the basics helps us identify the necessary steps for simplification.
When simplifying algebraic expressions, the goal is to find an equivalent expression that is easier to work with. This can involve combining like terms, factoring, or using algebraic identities (which is the focus of this problem).
The Difference of Squares
The difference of squares is a special algebraic identity that makes simplification straightforward.
It states that: \[a^2 - b^2 = (a + b)(a - b)\].
This means that if you have a product of two binomials where one binomial is the sum of two terms and the other is the difference of the same two terms, you can simplify it to the difference of their squares.
In our step-by-step solution, we recognize that \((7 + \sqrt{10})(7 - \sqrt{10})\) fits this pattern.
If we identify \(a\) as 7 and \(b\) as \(\sqrt{10}\), applying the difference of squares formula gives us:
\(7^2 - (\sqrt{10})^2\).
This is much simpler to work with.
It states that: \[a^2 - b^2 = (a + b)(a - b)\].
This means that if you have a product of two binomials where one binomial is the sum of two terms and the other is the difference of the same two terms, you can simplify it to the difference of their squares.
In our step-by-step solution, we recognize that \((7 + \sqrt{10})(7 - \sqrt{10})\) fits this pattern.
If we identify \(a\) as 7 and \(b\) as \(\sqrt{10}\), applying the difference of squares formula gives us:
\(7^2 - (\sqrt{10})^2\).
This is much simpler to work with.
Simplification Steps Made Easy
The simplification steps in our example involve several key ideas.
First, recognize the pattern of the difference of squares.
Then identify your \(a\) and \(b\).
Here are the steps of the simplification:
First, recognize the pattern of the difference of squares.
Then identify your \(a\) and \(b\).
Here are the steps of the simplification:
- Identify the given expression: \((7 + \sqrt{10})(7 - \sqrt{10})\).
- Recognize it as a difference of squares: \((a + b)(a - b) = a^2 - b^2\).
- Apply the formula: with \(a = 7\) and \(b = \sqrt{10}\), we have \(7^2 - (\sqrt{10})^2\).
- Simplify: \(7^2 = 49\) and \((\sqrt{10})^2 = 10\).
- Final answer: \(49 - 10 = 39\).
Other exercises in this chapter
Problem 292
In the following exercises, simplify. $$ (3-\sqrt{5})(3+\sqrt{5}) $$
View solution Problem 293
In the following exercises, simplify. $$ (10-\sqrt{3})(10+\sqrt{3}) $$
View solution Problem 296
In the following exercises, simplify. $$ (4+9 \sqrt{3})(4-9 \sqrt{3}) $$
View solution Problem 297
In the following exercises, simplify. $$ (1+8 \sqrt{2})(1-8 \sqrt{2}) $$
View solution