An intricately designed chalkboard with mathematical equations cleanly written on it. On the left part of the chalkboard the mathematical expression '81a^2' is prominently displayed, symbolizing a raised power of 81 by a variable 'a'. On the right side of the chalkboard, divorced from the first equation by slight space, is '5b^2' showcasing a different mathematical function involving the multiplication of the number '5' with the variable 'b' raised to the power of '2'. The chalkboard has a vintage wooden frame, and it is adorned with a chalk and an eraser on a ledge at the bottom.

Factor:

81a^2+5b^2

81a^2+5b^2

= 81a^2 - 5b^2i^2 , where i = √-1, thus i^2 = -1
= (9a + √5 bi)(9a - √5bi)

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An
2
+ 21n + 27 = 7

Why did the math book go to the comedy club? Because it wanted to find some factoring jokes! Alright, let's get down to business and factor 81a^2 + 5b^2. This expression can be factored as (9a + √5bi)(9a - √5bi). Wondering why there's an imaginary number involved? Well, it adds a bit of complexity to the equation and makes factoring much more interesting, don't you think?

To factor the expression 81a^2 + 5b^2, we will look for common factors among the two terms.

First, let's look at the coefficients: 81 and 5. They do not share any common factors.

Next, let's consider the variables: a^2 and b^2. These terms are not like terms because they have different variables.

Therefore, the expression 81a^2 + 5b^2 cannot be factored further over the integers.

To factor the expression 81a^2 + 5b^2, we can start by noticing that both terms are perfect squares.

First, let's look at the term 81a^2. We can rewrite it as (9a)^2, since (9a)^2 is equal to 81a^2.

Next, let's look at the term 5b^2. It is already a perfect square, so we don't need to do anything else with it.

Now, we can rewrite the expression as follows:

81a^2 + 5b^2
(9a)^2 + (sqrt(5)b)^2

Notice that we have a sum of two perfect squares: (9a)^2 and (sqrt(5)b)^2.

When we have a sum of two perfect squares, we can use the pattern (a^2 + b^2) = (a + b)(a - b) to factor them.

Applying this pattern to our expression, we get:

(9a + sqrt(5)b)(9a - sqrt(5)b)

So, we have factored the expression 81a^2 + 5b^2 as (9a + sqrt(5)b)(9a - sqrt(5)b).