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1.8 C) Multiplying Surds – Part 2
1.8 C) Multiplying Surds – Part 2
We are now going to have a look at a few more complex examples whereby we are multiplying two surds that have a number that is outside of the surd. Whenever we are multiplying surds that have numbers in front of them, we are going to get an answer that is in the form a√b. The value of a is obtained by multiplying the two numbers that are on the outside of the surds. The value of b is obtained by multiplying the numbers that are inside the surd.
Example 1
Complete the calculation below.
Complete the calculation below.
We will get an answer that is in the form a√b. a is obtained by multiplying the numbers that are on the outside of the surds; we multiply 3 by 2, which means that a 6. The next step is to find the value of b, which we obtain by multiplying the numbers that are inside of each of the surds; we multiply 5 by 7, which means that b is 35. Our surd is:
The final step is to check whether our surd is in its simplest form, which is done by checking to see if there are any square factors in the number that is inside the surd. There are no square factors in 35, thus meaning that our surd is in its simplest form.
Example 2
Complete the calculation below.
Complete the calculation below.
It is usually a good idea to check whether any of the surds that you are multiplying together can be simplified. This is because simplifying surds before we multiply will mean that we are multiplying numbers that are smaller, thus making our calculations easier. The easiest way to simplify surds is to find the highest square factor for the number that is inside the surd.
There are no square factors in 7, which means that we are unable to simplify the first surd. The second surd has 32 inside the surd. 16 is a square factor of 32 and this means that the second surd can be simplified (when we are simplifying the second surd, we need to be really carful with the 2 on the outside of the surd). The simplification process for the second surd is shown below.
There are no square factors in 7, which means that we are unable to simplify the first surd. The second surd has 32 inside the surd. 16 is a square factor of 32 and this means that the second surd can be simplified (when we are simplifying the second surd, we need to be really carful with the 2 on the outside of the surd). The simplification process for the second surd is shown below.
The calculation can be now written as:
The answer will be in the form a√b, where a is found by multiplying the numbers on the outside of the surds and b is found by multiplying the numbers that are on the inside of the surds. This means that a is found by multiplying 10 by 8, which means that a is 80. b is found by multiplying 7 by 2, which means that b is 14. Therefore, the answer is:
The final step is to check whether the surd can be simplified by seeing if there are any square factors in 14. There are no square factors in 14 and this means that the answer is in its simplest form.
We still would have been able to obtain the same final answer if we did not simplify the surds that we were multiplying at the start. The process will be exactly the same; a will be found by multiplying the numbers outside the surds and b will be found by multiplying the numbers inside the surds. Therefore, a is 20 (10 x 2) and b is 224. This would give us the answer:
We still would have been able to obtain the same final answer if we did not simplify the surds that we were multiplying at the start. The process will be exactly the same; a will be found by multiplying the numbers outside the surds and b will be found by multiplying the numbers inside the surds. Therefore, a is 20 (10 x 2) and b is 224. This would give us the answer:
The next step would be to check for square factors in the number that is inside the surd; we are looking for any square factors in 224. The highest square factor in 224 is 16.
You can see from both of these calculations that simplifying the surds before multiplying is considerable easier than simplifying the answer at the end. However, if you forget to simplify the surds at the start or prefer not to, it does not matter as you will still obtain the same final answer.