Video: Simplifying Numerical Expressions Using the Properties of Square Roots

Express (√7 + √3)² − √84 in its simplest form.

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Video Transcript

Express root seven plus root three all squared minus root 84 in its simplest form.

In order to answer this question, we will firstly simplify root seven plus root three all squared. Then, we will simplify root 84. Root seven plus root three all squared can be rewritten as root seven plus root three multiplied by root seven plus root three.

We can expand these two brackets using the FOIL method. Before we start, it is worth remembering some of the laws of surds. Root 𝑎 multiplied by root 𝑎 is equal to 𝑎. Also root 𝑎 multiplied by root 𝑏 is equal to the root of 𝑎 multiplied by 𝑏. And finally, root 𝑎 plus root 𝑎 is equal to two root 𝑎.

Going back to our question, root seven multiplied by root seven is equal to seven. Multiplying the outside terms root seven and root three gives us root 21, as seven multiplied by three equals 21. Multiplying the inside terms also gives us root 21. And finally, multiplying the last terms gives us three. Root three multiplied by root three is equal to three. Collecting the like terms gives us 10 plus two root 21.

Root seven plus root three all squared is equal to 10 plus two root 21.

Our next step is to simplify root 84. In order to simplify root 84, we need to find a square number greater than one that divides exactly into 84 or is a factor of 84. Four multiplied by 21 is equal to 84. Therefore, 84 can be rewritten as root four multiplied by root 21. The square root of four is equal to two.

Therefore, root 84 is equal to two root 21.

We now know that root seven plus root three all squared minus root 84 is equal to 10 plus two root 21 minus two root 21. Two root 21 minus two root 21 is equal to zero. This means that our answer is 10.

Root seven plus root three all squared minus root 84 in its simplest form is equal to 10.

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