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1.8 F) Expanding Double Brackets with Surds
1.8 F) Expanding Double Brackets with Surds
We are now going to work through a few examples of expanding double brackets that contain surds. Whenever we are expanding double brackets, we need to make sure that we multiply every term in each bracket by all of the terms in the other bracket. Click here to be taken through to the section on multiplying out double brackets because it is best to master the multiplying out of double brackets with numbers and unknowns before trying to multiply out brackets containing surds.
Example 1
Expand the brackets below:
Expand the brackets below:
I am going to use the FOIL technique (First, Outside, Inside, Last).
Therefore, I will start by multiplying the first terms in each of the brackets, which is 5 multiplied by 2 and this is 10.
Therefore, I will start by multiplying the first terms in each of the brackets, which is 5 multiplied by 2 and this is 10.
The next step is to multiply the outside terms, which is 5 multiplied by √3. The best way to think about multiplying surds by a number is to imagine the surd as an unknown. Therefore, we are multiplying 5 by an unknown (where the unknown is √3). This means that we are going to get 5 unknowns. Our unknown was √3, which means that the answer to 5 multiplied by √3 is 5√3.
We now multiply the inside terms, which is √3 multiplied by 2. Like before, we should treat the surd as an unknown, which means that we are multiplying an unknown by 2. This will give us 2 unknowns meaning that the answer is going to be 2√3.
The final multiplication is of the last terms in each of the brackets, which is √3 multiplied by √3. When we multiply a surd by the same surd, the answer is the number that is inside the surds (the rule is: √a x √a = a). Therefore, √3 multiplied by √3 is 3.
The final step is to simplify our answer by combining terms that are the same. There are two different types of terms in the answer above; numbers and √3’s.
The two numbers combined give us 13 (10 + 3). We now need to combine the √3’s. There are two different √3 terms; 5√3 and 2√3. The best way to combine surds that are same is to think of the surds as unknowns and I am going to let √3 equal x.
It now a simple case of adding, which gives us 7x. The final step is to replace x with √3. This means that 5√3 + 2√3 is 7√3.
We now combine the 7√3 with the 13 to give us the final answer:
We now combine the 7√3 with the 13 to give us the final answer:
Example 2
Expand the bracket below:
Expand the bracket below:
I am going to use the FOIL technique (First, Outside, Inside, Last).
The first step in the FOIL technique is to multiply the first terms in each bracket. This means that we will be multiplying 2 and 6, which gives us 12.
The first step in the FOIL technique is to multiply the first terms in each bracket. This means that we will be multiplying 2 and 6, which gives us 12.
The next step is to multiply the outside terms, which are 2 and -√2. We are multiplying a positive by a negative, which means that our answer is going to be negative. The best way to imagine multiplying a number by a surd is to imagine that the surd is an “unknown”. Therefore, you can imagine that you are multiplying an unknown by 2, which mean that you will have 2 “unknowns”. We can now replace the unknown with the surd and combine it with the fact that the answer will be negative. This means that 2 multiplied by -√2 is -2√2
We now multiply the inside terms, which are √7 and 6. This gives us 6√7.
The final step is to multiply the last terms in each of the brackets, which means that we will be multiplying √7 by -√2. The answer is going to be negative because we are multiplying a positive by a negative. Whenever we are multiplying two surds, we multiply the numbers that are inside each of the surds; the rule is that √a x √b = √ab. Therefore, the number inside our surd is going to be 14. This means that √7 x -√2 is -√14.
We now need to make sure that our answer is given in its simplest form.
We are only able to collect surds if the numbers inside the surds are the same. There are currently no numbers inside the surds that are the same and there are no surds that can be simplified (there are no square factors in 2, 7 or 14). This means that our answer is in its simplest form.
We are only able to collect surds if the numbers inside the surds are the same. There are currently no numbers inside the surds that are the same and there are no surds that can be simplified (there are no square factors in 2, 7 or 14). This means that our answer is in its simplest form.