Multiply this radical expression.?

Question

Multiply:
(√[3] + 4√[5])(2√[3] – √[5])

Please show steps as I can not seem to figure this problem out. You do not have to give the final answer, just help me get on track with the way to work this problem. Thanks all in advance.

Answer 1

Rule of thumb when multiplying radicals is that you can always multiply two square roots into one square root. For example,
sqrt(5)sqrt(3) = sqrt(15). Let's do this example.

[sqrt(3) + 4sqrt(5)] [2sqrt(3) - sqrt(5)]

Use FOIL as you would a regular binomial multiplied by a regular binomial. I'll show you the exact details so you can see.

[sqrt(3)][2sqrt(3)] - [sqrt(3)][sqrt(5)] + [4sqrt(5)][2sqrt(3)] -
[4sqrt(5)][sqrt(5)]

Note that the square root of a number times the square root of the same number is itself, i.e. sqrt(3)sqrt(3) = 3. Also, we can multiply the constants (coefficients of the square roots) normally.
Let's multiply the constants first.

3sqrt(3)sqrt(3) - sqrt(3)sqrt(5) + 8sqrt(5)sqrt(3) - 4sqrt(5)sqrt(5)

Note that we can rearrange sqrt(5)sqrt(3) as sqrt(3)sqrt(5), since multiplication is commutative.

3(3) - sqrt(3)sqrt(5) + 8sqrt(3)sqrt(5) - 4(5)

We can actually group the middle terms as if we were doing
-x + 8x.

3(3) + 7sqrt(3)sqrt(5) - 4(5)
9 + 7sqrt(3)sqrt(5) - 20
-11 + 7sqrt(3)sqrt(5)

Finally, we can merge sqrt(3)sqrt(5) into sqrt(15).

-11 + 7sqrt(15)

Answer 2

If it helps any replace the √[3] by x and the √[5] by y, then multiply your expression. Then replace the x and y with the proper values. Of course √[5] X √[5] is 5 and the same for the √3 times itself. And √5 X √3 would be √15 . Does this help?

Answer 3

(√[3] + 4√[5])(2√[3] – √[5])=
= √[3]*2√[3] + 4√[5]*2√[3] - √[3]*√[5] - 4√[5]*√[5] =
= 2*3 + 8√[15] - √[15] -4*5 =
= 6 + 7√[15] - 20 =
= 7√[15] -14 (final result)

Answer 4

Use the distributive property to multiply the first term in each bracket (sqrt3 * 2squrt3), then add that to the product of the second two terms in each. Then multiply the inside terms and add them, then multiply the two outside ones and add them.

Note: adding a negative is actually subtracting. keep this straight.