How do you simplify (3sqrt7 - 6sqrt5)(4sqrt7 + 8sqrt5)(3√7−6√5)(4√7+8√5)?

1 Answer
Roy Ø.
May 18, 2018

(3sqrt7-6sqrt5)(4sqrt7+8sqrt5)=-156(3√7−6√5)(4√7+8√5)=−156

Explanation:

Simplifying would mean getting rid of as many square roots as possible. We should, therefore, multiply it out.

First we note that 3 is common in the first parenthesis, and 4 is common in the second. Therefore:

(3sqrt7-6sqrt5)(4sqrt7+8sqrt5)(3√7−6√5)(4√7+8√5)
=3(sqrt7-2sqrt5)*4(sqrt7+2sqrt5)=3(√7−2√5)⋅4(√7+2√5)

You should know that (a-b)(a+b)=a^2-b^2(a−b)(a+b)=a2−b2,
which means that (sqrt7-2sqrt5)(sqrt7+2sqrt5)=7-2^2*5=-13(√7−2√5)(√7+2√5)=7−22⋅5=−13

Therefore =3(sqrt7-2sqrt5)*4(sqrt7+2sqrt5)=-3*4*13=-156=3(√7−2√5)⋅4(√7+2√5)=−3⋅4⋅13=−156

Ergo (3sqrt7-6sqrt5)(4sqrt7+8sqrt5)=-156(3√7−6√5)(4√7+8√5)=−156

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