- #1
mathdad
- 1,283
- 1
How does (2sqrt{7})(sqrt{8 - 2sqrt{7}) become
2(7 - sqrt{7})?
2(7 - sqrt{7})?
MarkFL said:Well, we may write:
\(\displaystyle 8-2\sqrt{7}=7-2\sqrt{7}+1=(\sqrt{7}-1)^2\)
And so:
\(\displaystyle \sqrt{8-2\sqrt{7}}=\sqrt{(\sqrt{7}-1)^2}=\sqrt{7}-1\)
Thus:
\(\displaystyle 2\sqrt{7}\sqrt{8-2\sqrt{7}}=2\sqrt{7}(\sqrt{7}-1)=2(7-\sqrt{7})\)
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}?
MarkFL said:Well if you did that, you would have:
\(\displaystyle 2\sqrt{7}\sqrt{8-2\sqrt{7}}=2\sqrt{56-14\sqrt{7}}=2\sqrt{49-14\sqrt{7}+7}=2\sqrt{(7-\sqrt{7})^2}=2(7-\sqrt{7})\)
RTCNTC said:I get it except for 49 in the radical. Where did 49 come from?
MarkFL said:\(\displaystyle 56=49+7\)
Radical x Radical refers to the mathematical operation of multiplying two radical expressions together.
To simplify "Radical x Radical" expressions, you can use the product rule of radicals, which states that the product of two radicals with the same index can be simplified by multiplying their radicands (the numbers under the radical symbol) together.
Yes, "Radical x Radical" expressions can have different indices, but they cannot be simplified using the product rule. In this case, you can convert one or both radicals to have the same index before multiplying.
"Radical x Radical" expressions are commonly used in geometry and physics to solve problems involving area, volume, and distance. They are also used in electrical engineering to calculate the impedance of circuits.
When multiplying two "Radical x Radical" expressions, you should always check for like terms and combine them, if possible. Also, if the radicals have variables, you should simplify the expression by combining like terms and reducing the radicals, if possible.