How to Simplify a Radicand by Breaking it into Two Parts?

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Discussion Overview

The discussion revolves around the simplification of a radicand involving the expression (2√7)(√(8 - 2√7)). Participants explore different methods to simplify this expression, including breaking down the radicand and using multiplication rules for radicals. The focus is on mathematical reasoning and clarification of steps involved in the simplification process.

Discussion Character

  • Mathematical reasoning
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant asks how (2√7)(√(8 - 2√7)) simplifies to 2(7 - √7).
  • Another participant proposes that 8 - 2√7 can be rewritten as (√7 - 1)², leading to the conclusion that √(8 - 2√7) = √7 - 1.
  • A similar point is reiterated by another participant, confirming the simplification process and arriving at the same result.
  • One participant suggests an alternative approach by multiplying the radicals directly using the rule √a * √b = √(ab), which also leads to the same final expression.
  • A participant questions the origin of the number 49 in the radical during the multiplication process.
  • Another participant clarifies that 56 can be expressed as 49 + 7, explaining the breakdown of the radicand.

Areas of Agreement / Disagreement

Participants generally agree on the simplification steps leading to the expression 2(7 - √7) but explore different methods to arrive at this conclusion. There is no explicit disagreement, but the discussion includes various approaches to the problem.

Contextual Notes

The discussion involves assumptions about the properties of radicals and the manipulation of expressions, which may not be universally applicable without further context.

mathdad
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How does (2sqrt{7})(sqrt{8 - 2sqrt{7}) become
2(7 - sqrt{7})?
 
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Well, we may write:

$$8-2\sqrt{7}=7-2\sqrt{7}+1=(\sqrt{7}-1)^2$$

And so:

$$\sqrt{8-2\sqrt{7}}=\sqrt{(\sqrt{7}-1)^2}=\sqrt{7}-1$$

Thus:

$$2\sqrt{7}\sqrt{8-2\sqrt{7}}=2\sqrt{7}(\sqrt{7}-1)=2(7-\sqrt{7})$$
 
MarkFL said:
Well, we may write:

$$8-2\sqrt{7}=7-2\sqrt{7}+1=(\sqrt{7}-1)^2$$

And so:

$$\sqrt{8-2\sqrt{7}}=\sqrt{(\sqrt{7}-1)^2}=\sqrt{7}-1$$

Thus:

$$2\sqrt{7}\sqrt{8-2\sqrt{7}}=2\sqrt{7}(\sqrt{7}-1)=2(7-\sqrt{7})$$

Nicely done as always. What if I decided to multiply the two given radicals using the rule sqrt{a}*sqrt{b} = sqrt{ab}?
 
RTCNTC said:
Nicely done as always. What if I decided to multiply the two given radicals using the rule sqrt{a}*sqrt{b} = sqrt{ab}?

Well if you did that, you would have:

$$2\sqrt{7}\sqrt{8-2\sqrt{7}}=2\sqrt{56-14\sqrt{7}}=2\sqrt{49-14\sqrt{7}+7}=2\sqrt{(7-\sqrt{7})^2}=2(7-\sqrt{7})$$
 
MarkFL said:
Well if you did that, you would have:

$$2\sqrt{7}\sqrt{8-2\sqrt{7}}=2\sqrt{56-14\sqrt{7}}=2\sqrt{49-14\sqrt{7}+7}=2\sqrt{(7-\sqrt{7})^2}=2(7-\sqrt{7})$$

I get it except for 49 in the radical. Where did 49 come from?
 
RTCNTC said:
I get it except for 49 in the radical. Where did 49 come from?

$$56=49+7$$
 
MarkFL said:
$$56=49+7$$

I understand now. You broke 56 into two parts. This is a fine "trick" or step to simplify the radicand.
 

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