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1. Find the value of  $$\sqrt{a}\cdot\sqrt{a+6}\cdot\sqrt{b}\cdot\sqrt{b+6} for (a,b)=(7,91)$$.

2. What is the smallest positive integer $$k$$ such that  $$\sqrt{98 \cdot k}$$ is an integer?

3. (Just a hint please, I don't know why but I got 192) $$Simplify (\sqrt{3}+\sqrt{24}+\sqrt{192})^3$$

Thanks.

Aug 5, 2018
edited by Guest  Aug 5, 2018

#1
+1

3)

Simplify the following:
(3^(1/3) + 24^(1/3) + 192^(1/3))^3

24^(1/3) = (2^3×3)^(1/3) = 2 3^(1/3):
(3^(1/3) + 2 3^(1/3) + 192^(1/3))^3

192^(1/3) = (2^6×3)^(1/3) = 2^2 3^(1/3):
(3^(1/3) + 2 3^(1/3) + 2^2 3^(1/3))^3

2^2 = 4:
(3^(1/3) + 2 3^(1/3) + 4 3^(1/3))^3

Add like terms. 3^(1/3) + 2 3^(1/3) + 4 3^(1/3) = 7 3^(1/3):
(7 3^(1/3))^3

Multiply each exponent in 7 3^(1/3) by 3:
7^3×3^(3/3)

3/3 = 1:
7^3×3

7^3 = 7×7^2:
7×7^2×3

7^2 = 49:
7×49×3

7×49 = 343:
343×3

343×3 = 1029:

=1029

2)

Since 98 = 2 x 7^2, it, therefore, follows that if you multiply 98 by: 2^3 x 7^2 =392 you will have:

(98 x 392)^1/4 =14. Therefore k=392

1)

Sqrt(7) x Sqrt(7 + 6) x sqrt(91) x sqrt(91 + 6) =

Simplify the following:
sqrt(7) sqrt(7 + 6) sqrt(91) sqrt(91 + 6)

91 + 6 = 97:
sqrt(7) sqrt(7 + 6) sqrt(91) sqrt(97 )

7 + 6 = 13:
sqrt(7) sqrt(13 ) sqrt(91) sqrt(97)

sqrt(7) sqrt(13) sqrt(91) sqrt(97) = sqrt(7×13×91×97):
sqrt(7×13×91×97)

7×13 = 91:
sqrt(91×91×97)

91×91 = 8281:
sqrt(8281×97)

8281×97 = 803257:
sqrt(803257 )

sqrt(803257) = sqrt(91^2×97) = 91 sqrt(97):

=91 sqrt(97)=896.246

Aug 5, 2018
#2
+1

thanks for explaining in details! Guest Aug 5, 2018