Please help with question 6, 7, 8, and 9 about rational and irrational numbers
1 answer:
6. When we multiply two surdic integers, we can leave it into one big surd.
For instance, if we have:
, we can simplify it down to one uniform square root sign, 
.
So, we can say:
, which is a rational root, because the square root of 36 can be converted into an integer, namely 6 and -6.
Hence, it is a rational root.
7. The summation, on the other hand, cannot be simplified any further, so it will stay as irrational. This is because √2 is already an irrational root, and adding it onto a number that is rational will stay as irrational.
8. Once again, the summation cannot be simplified. We can only simplify by factoring like terms. Hence, this must be irrational.
9. Even if it is in fractional notation, √5 can never be a rational number because no two integers when multiplied by itself will give 5. So, even if it is written in fractional form, they cannot be rational if there is an already irrational term in it.
For instance, if we have:
So, we can say:
Hence, it is a rational root.
7. The summation, on the other hand, cannot be simplified any further, so it will stay as irrational. This is because √2 is already an irrational root, and adding it onto a number that is rational will stay as irrational.
8. Once again, the summation cannot be simplified. We can only simplify by factoring like terms. Hence, this must be irrational.
9. Even if it is in fractional notation, √5 can never be a rational number because no two integers when multiplied by itself will give 5. So, even if it is written in fractional form, they cannot be rational if there is an already irrational term in it.
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When we approach limits, we are finding values that are infinitesimally approaching this x-value. Essentially, we consider the approximate location that this root or limit appears. This is essential when it comes to taking Calculus, and finding the limit or rate of change of a function.
When we are attempting limits questions, there are several tests we attempt first.
1. Evaluate the limit by substituting the value of the x-value as it approaches the value (direct evaluation of a limit)
2. Rearrangement of the function, such that we can evaluate the limit.
3. (TRIGONOMETRIC PROPERTIES)
4. Using L'Hopital's Rule for indeterminate limits, such as 0/0, -infinity/infinity, or infinity/infinity.
For example:
1)
We can do this using the first and second method.
<em>Method 1: Direct evaluation:</em>
Substitute x = 0 to the function.
<em>Method 2: Rearranging the function
</em>
We can see that x - 25 can be rewritten as: (√x - 5)(√x + 5)
By rewriting it in this form, the top will cancel with the bottom easily, and our limit comes out the same.
Every example works exactly the same way, and by remembering these criteria, every limit question should come out pretty naturally.
When we are attempting limits questions, there are several tests we attempt first.
1. Evaluate the limit by substituting the value of the x-value as it approaches the value (direct evaluation of a limit)
2. Rearrangement of the function, such that we can evaluate the limit.
3. (TRIGONOMETRIC PROPERTIES)
4. Using L'Hopital's Rule for indeterminate limits, such as 0/0, -infinity/infinity, or infinity/infinity.
For example:
1)
We can do this using the first and second method.
<em>Method 1: Direct evaluation:</em>
Substitute x = 0 to the function.
<em>Method 2: Rearranging the function
</em>
We can see that x - 25 can be rewritten as: (√x - 5)(√x + 5)
By rewriting it in this form, the top will cancel with the bottom easily, and our limit comes out the same.
Every example works exactly the same way, and by remembering these criteria, every limit question should come out pretty naturally.