This week in Pre-Calc 11, I learned how to convert irrational trinomials to perfect square trinomials and apply them to solve quadratic equations. However, before we can start solving quadratic equations with perfect square trinomials we must first understand how to recognize and complete them.

Let’s start with recognizing a perfect square trinomial. A sure fire way to to identify a perfect square trinomial is by looking at it’s first and third term. In a perfect square trinomial the first and third term are always a perfect square think of 4, 9, 16 25.. Ect, and half of the second term is the product of the the root of first and third multiplied.

Now, the second skill you need to learn to solve quadratic equations with perfect squares is completing square trinomials. You can do this by dividing the second term by two then squaring it  to find the missing term.

Now that you have learned the skills needed let’s apply what we learnt into converting an irrational trinomial and turn it into a perfect square trinomial in order to isolate X. In our example here we can clearly tell this equation as the third term is not a perfect square. So our first step is to find out what will complete our perfect square.

As we learned earlier we can find our missing term by dividing the second term by two then squaring it to find the missing term. So that’s exactly what we are gonna do here.

And now we know our missing term. We add the amount needed to get our third term to 9 and what we do on one side of the equation we must do on the other side.

Now we convert our perfect trinomial back into a perfect binomial.

We get rid of the square by square rooting both side.

Now we isolate X by subtracting 3 leaving us with our answer.

This week in Precalculus 11, I learned how to rationalize denominators. Rationalizing denominators is crucial when dealing with fractions containing radicals in the denominator. It helps in removing the radical from the denominator, making it much easier to work with. So here’s how we do it.

In this example we are dealing with a lone root, and we can tell that the radical in the denominator is not rational because it’s not a perfect square.

When dealing with roots we simply multiply the denominator by itself, and what we do to the bottom we do to the top so also we multiply 6 with root 6.

Then we simply divide leaving us with root 6.

Now let’s try another problem, in this example we are dealing with a binomial as the denominator.

We can rationalize it by multiplying both the top and the cotton by a conjugate of the binomial.

We put our term together.

Leaving us with root 5 + 1

And now you know how to rationalize denominator.

This week in Precalculus 11, I learned about multiplying mixed radicals. It’s an important skill to learn as in practice you are most likely to encounter equations where you have to multiply mixed radicals.One very important rule to remember while multiplying mixed radicals is to multiply the coefficient and the radicand separately.

In our example here our coefficient is 2 and our radicand is root 3 multiplied by 2 root 9.

Now we multiply separately giving us the answer of 4 root 27.

Lastly, further simplify your radical by factoring, and tada we did it!

Now let’s try a harder question. First let’s apply 2 root 3 into the brackets.

Here it is now spread out.

Now if possible simplify the equation, but in this case we cannot simplify nor subtract because they don’t have the same radicand.

Now you know what to do when you encounter an equation with mixed radicals!

This week in Precalc 11 I learned how to divide radicals, it is important to know especially when you are simplifying or solving equations with radicals.

Here’s how you do it! In others for us to divide radicals first we must check if the radicals share the same index. In our example 2 (square) is our index. Then divide the radicand by the radicand and the coefficient by the coefficient.

Example:

Some of you might be confused where the ones come from but remembers, there are always invisible 1s (coefficient) in-front of your radical. This is just and example to remind you.

There are also in some situations where your if the denominator will contain an irrational number under the radical. What you then do is simply multiply the both the numerator and the denominator by the denominator. This will then convert it from irrational to rational.

Thank you again for coming back to another blog post and see ya again next week!

In the first week back of semester 2, we relearned how to do radicals! Radicals are made up of 3 parts, the index (n), the radicand (x), the radical symbol (√) and sometimes the coefficient .

Explanation for each part:

Index:

• Shows you how many times you need to multiply the same number to get the radicand
• Has to be a Natural number.
• If there is no number in the index assume it is represented by an invisible 2.

Radicand:

• The radicand is the number inside the radical symbol. It’s the value for which you’re finding the root.

Radical:

• The radical symbol represents the root operation.

Coefficient:

• A numerical factor that multiplies the entire radical expression.
• It should be applied after the root

  So now that we have a basic understanding of the parts of a radical lets try an example.

  

  In this equation, the coefficient is 5, the index is 3 (a cube root), and the radicand is 27. It’s asking, “What number, when multiplied 3 time , gives you 27”

  Well we know that 9×3=27 and 3×3=9 this mean the cube root of 27 is

  

  That leaves us with this 5×3 and we simply just multiply the coefficient with the root number to get our answer of 15, well wasn’t that simple?

  

Happy Chinese New Year, in this weeks lesson we reviewed the radical laws! Consisting of the zero exponent law, product law, quotient law, power law and the negative exponent law. These laws play an important role t in simplifying difficult equations with powers.

Zero Exponent Laws:

When a number is to the power of 0, it equals 1. This is because any number to the power of zero is equal to the multiplicative identity, which is 1. Citation – (https://byjus.com/question-answer/why-any-number-power-0-1/)

Product Law:

When multiplying two powers, you add their exponents together but only if they have the same base. (base = x)

Quotient Law:

When dividing two powers, you subtract the exponent of the denominator from the exponent of the numerator.

Power Law:

When a power is raised to another power, you can multiply the exponents together to simplify the equation!

Negative Exponent:

A negative exponent indicates the reciprocal of the base raised to the positive exponent.