Problem 52
Question
Find each of the following products. $$ \sqrt{h+1} \sqrt{h-1} $$
Step-by-Step Solution
Verified Answer
Answer: The product of the square roots is $\sqrt{h^2-1}$.
1Step 1: Write down the given expression
Write down the given product of square roots:
$$
\sqrt{h+1} \sqrt{h-1}
$$
2Step 2: Apply the product property of square roots
Use the product property of square roots, \(\sqrt{a} \sqrt{b} = \sqrt{ab}\), to combine the two terms:
$$
\sqrt{(h+1)(h-1)}
$$
3Step 3: Simplify the expression inside the square root
Recognize that the expression inside the square root is a difference of squares and can be simplified:
$$
\sqrt{h^2 - 1^2}
$$
4Step 4: Write the final answer
After simplifying the expression, we find the product of the given square roots:
$$
\sqrt{h^2-1}
$$
Key Concepts
Difference of SquaresSimplifying ExpressionsProduct Property of Square Roots
Difference of Squares
The difference of squares is a mathematical pattern that occurs when you have two terms that are squares being subtracted from each other. This pattern is expressed as \[(a^2 - b^2) = (a + b)(a - b)\],which means that the difference of two squares can be factored into the product of a sum and a difference.
In the exercise, notice how \((h+1)(h-1)\) fits the structure of the difference of squares. Here, \(h\) is the value for \(a\) and \(1\) is the value for \(b\).
When multiplied together, you get\(h^2 - 1\), thereby simplifying the expression inside the square root. Embracing this pattern makes solving similar problems quicker. You don't have to resort to lengthy multiplication, as recognizing this pattern let’s you swiftly rewrite products as simpler expressions.
In the exercise, notice how \((h+1)(h-1)\) fits the structure of the difference of squares. Here, \(h\) is the value for \(a\) and \(1\) is the value for \(b\).
When multiplied together, you get\(h^2 - 1\), thereby simplifying the expression inside the square root. Embracing this pattern makes solving similar problems quicker. You don't have to resort to lengthy multiplication, as recognizing this pattern let’s you swiftly rewrite products as simpler expressions.
Simplifying Expressions
When simplifying expressions, the goal is to reduce them to their simplest form. In the exercise, after using the product property, we have the expression \(\sqrt{(h+1)(h-1)}\).
To simplify, we first recognize the difference of squares. This gives us \(h^2 - 1\), which is already considerably simpler than the original product form.
By expressing as \(h^2 - 1\), we are free of unnecessary parentheses, and it's straightforward to compute the square root if needed in future steps.
Simplifying expressions not only aids in obtaining a cleaner result but also makes further mathematical operations easier to manage. It’s important to look for patterns like difference of squares that enable such simplification.
To simplify, we first recognize the difference of squares. This gives us \(h^2 - 1\), which is already considerably simpler than the original product form.
By expressing as \(h^2 - 1\), we are free of unnecessary parentheses, and it's straightforward to compute the square root if needed in future steps.
Simplifying expressions not only aids in obtaining a cleaner result but also makes further mathematical operations easier to manage. It’s important to look for patterns like difference of squares that enable such simplification.
Product Property of Square Roots
The product property of square roots is a useful tool that states \(\sqrt{a} \sqrt{b} = \sqrt{ab}\).
Using this property allows us to combine square roots into a single square root, simplifying calculations especially when the numbers inside the square roots relate to each other, like in this exercise.
When given the expression \(\sqrt{h+1} \sqrt{h-1}\), the product property licenses us to write it as \(\sqrt{(h+1)(h-1)}\).
Once combined, this single square root is easier to manage and can be further simplified using tools like the difference of squares.
The product property is invaluable for simplifying expressions and solving problems efficiently, particularly when dealing with longer series of multiplicative expressions under square roots.
Using this property allows us to combine square roots into a single square root, simplifying calculations especially when the numbers inside the square roots relate to each other, like in this exercise.
When given the expression \(\sqrt{h+1} \sqrt{h-1}\), the product property licenses us to write it as \(\sqrt{(h+1)(h-1)}\).
Once combined, this single square root is easier to manage and can be further simplified using tools like the difference of squares.
The product property is invaluable for simplifying expressions and solving problems efficiently, particularly when dealing with longer series of multiplicative expressions under square roots.
Other exercises in this chapter
Problem 52
Simplify each expression by performing the indicated operation. $$ (1+\sqrt{3})^{2} $$
View solution Problem 52
For the following problems, simplify each of the radical expressions. $$ \sqrt{r^{2}} $$
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For the following problems, simplify each expressions. $$ \frac{\sqrt{r+1}}{\sqrt{r-1}} $$
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Simplify each expression by performing the indicated operation. $$ (3+\sqrt{5})^{2} $$
View solution