Video: Simplifying Numerical Expressions Using the Properties of Conjugates of Irrational Numbers

Find the value of (√21 + √13)² (√21 − √13)².

05:33

Video Transcript

Find the value of root 21 plus root 13 all squared multiplied by root 21 minus root 13 all squared.

In order to solve this problem, we’ll use the properties of conjugates of irrational numbers and the following formula. Multiplying root 𝑎 plus root 𝑏 all squared by its conjugate root 𝑎 minus root 𝑏 all squared gives us 𝑎 squared minus two 𝑎𝑏 plus 𝑏 squared.

In our example, 𝑎 is equal to 21 and 𝑏 is equal to 13. Substituting in these values gives us 21 squared minus two multiplied by 21 multiplied by 13 plus 13 squared. 21 squared is calculated by multiplying 21 by 21. 21 multiplied by 20 is 420 and 21 multiplied by one is equal to 21. Adding these two numbers gives us 441. Therefore, 21 squared is equal to 441.

Our next step is to multiply two by 21 by 13. Two multiplied by 21 is equal to 42. So we need to multiply 42 by 13. 42 multiplied by 10 is 420 and 42 multiplied by three is 126. Adding these two numbers gives us 546. Therefore, two multiplied by 21 multiplied by 13 is equal to 546. Finally, 13 squared is equal to 169.

Grouping the positive terms gives us 610 as 441 plus 169 is equal to 610. Subtracting 546 from this gives us an answer of 64. This means that the root of 21 plus the root of 13 all squared multiplied by the root of 21 minus the root of 13 all squared is equal to 64.

An alternative method would’ve been to have expanded the two brackets separately and then multiplied our two expressions. Root 21 plus root 13 all squared is the same as root 21 plus root 13 multiplied by root 21 plus root 13. This can be expanded and simplified using the FOIL method.

Multiplying the first terms root 21 by root 21 gives us 21. Multiplying the outside terms gives us root 273 as 21 multiplied by 13 is 273. Multiplying the inside terms also gives us root of 273. Finally, multiplying the last terms gives us 13 as root 13 times root 13 is equal to 13. Simplifying this by collecting the like terms gives us 34 plus two root 273.

21 plus 13 is equal to 34 and root 273 plus root 273 is equal to two root 273. We can expand root 21 minus root 13 all squared in exactly the same way. Using the FOIL method this time gives us 21 minus root 273 minus root 273 plus 13, giving us a simplified form of 34 minus two root 273.

We’ve, therefore, worked out root 21 plus root 13 all squared and root 21 minus root 13 all squared.

Our next step is to multiply these two terms together. We could once again expand these brackets using the FOIL method. However, we might notice that this is the difference of two squares: 34 plus two root 273 and 34 minus two root 273.

This means that the outside and inside parts will cancel. We only need to multiply the first terms and the last terms. 34 multiplied by 34 or 34 squared is 1156. Two root 273 multiplied by two root 273 is equal to four multiplied by 273. This is equal to 1092. Subtracting this from 1156 once again gives us the answer 64.

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