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Summary

The video explains how to simplify nested radicals by rewriting the expression inside the outer square root as a perfect square:

• Addition pattern: [ \sqrt{a+b+2\sqrt{ab}}=\sqrt a+\sqrt b ]

• Subtraction pattern: [ \sqrt{a+b-2\sqrt{ab}}=\sqrt a-\sqrt b, \qquad a\ge b\ge 0. ]

The key is to find two nonnegative numbers whose sum matches the non-radical term and whose product matches the quantity under the inner radical. If the inner radical’s coefficient is not (2), first rewrite the expression to fit one of these patterns.

Method

1. Identify whether there is a plus or minus before the inner radical.
2. Rewrite the expression in the form (a+b\pm2\sqrt{ab}).
3. Find (a) and (b) by checking factor pairs:
   • Their sum must equal the non-radical term.
   • Their product must equal the quantity under the inner radical.
4. Apply the appropriate identity.
5. Simplify any resulting radicals, such as (\sqrt{25}=5) or (\sqrt{8}=2\sqrt{2}).

Examples

The examples simplify as follows:

• (\sqrt{12+2\sqrt{20}}=\sqrt{10}+\sqrt{2})
• (\sqrt{28-2\sqrt{75}}=5-\sqrt{3})
• (\sqrt{15+4\sqrt{14}}=2\sqrt{2}+\sqrt{7})
• (\sqrt{20-6\sqrt{11}}=\sqrt{11}-3)
• (\sqrt{28+5\sqrt{12}}=5+\sqrt{3})
• (\sqrt{26-3\sqrt{68}}=\sqrt{17}-3)
• (\sqrt{15+2\sqrt{26}}=\sqrt{13}+\sqrt{2})
• (\sqrt{46-2\sqrt{45}}=3\sqrt{5}-1)
• (\sqrt{16+4\sqrt{7}}=\sqrt{14}+\sqrt{2})

A final example uses a preliminary scaling step to fit the pattern:

[ \sqrt{7-\sqrt{33}}=\frac{\sqrt{22}-\sqrt{6}}{2}. ]

Some example numbering and wording in the autogenerated subtitles appear inconsistent, but the method and results above reflect the calculations presented.

Speaker or Source

A single instructor/narrator from Bimbel Jakarta Timur presents the lesson. No other speakers are clearly featured.