V = Radical
The expression:
V2(V5-12V3)-V3(V8-2V30) can be written in simplest form aVb-cVd, where a, b, c ,d are all positive integers. The value of a + b + c + d is ________.
So after solving all this out, I arrived at 7V10 - 14V6. That - is beside the 14, and Im only supposed to have positive integers, so does that negative sign mean anything at all *question mark* --Sorry accidentally pressed this button on keyboard, now Im getting all these french letters, so cant paste a question mark.
what the negative sign means is c is negative.
But all a,b,c and d are supposed to be positive integers, and I looked over this again, and its supposed to simplify into aVb-cVd, so that means that the negative sign is supposed to be there, but the integer C is not negative.
√2(√5-12√3)-√3(√8-2√30)
= √10 - 12√6 - √24 + 2√90
= √10 - 12√6 - 2√6 + 6√10
= 7√10 - 14√6
which is what you got.
comparing this to a√b - c√d
a=7 , b = 10 , c = -14, and d = 6
so
a+b+c+d = 7+10-14+6 = 9
Trust yourself!
To simplify the expression V2(V5-12V3)-V3(V8-2V30), let's break it down step by step:
1. Start with the expression V2(V5-12V3)-V3(V8-2V30).
2. Simplify each term inside the parentheses:
- V5 can be simplified as 5V1 (since the index is 5).
- V3 can be simplified as 3V1 (since the index is 3).
- V8 can be simplified as 2V2 (since the index is 8).
- V30 can be simplified as 5V6 (since the index is 30).
3. Apply the product of radicals rule: Vn * Vm = V(n+m).
- V2 * 5V1 = 10V1
- V2 * (-12V3) = -12V5
- V3 * 2V2 = 2V5
- V3 * (-2V6) = -2V9
4. Substitute the simplified terms back into the original expression:
10V1 - 12V5 - 2V5 - 2V9
5. Simplify further by combining like terms. Like terms have the same radical part (Vn) and can be added or subtracted:
-12V5 - 2V5 = -14V5
6. Rewrite the expression in simplest form:
10V1 - 14V5 - 2V9
Now, to answer your question about the negative sign: the negative sign in front of a radical signifies the negative value of that radical. In the expression -14V5, the negative sign indicates that the value of the radical is negative (-14V5 is equivalent to -14 times the square root of 5).
As for the positive integers, they refer to the coefficients (numbers outside the radicals). In this case, a = 10, b = 1 (since there is no number outside V1, it is considered as 1), c = 14, and d = 5.
To find the sum a + b + c + d:
a + b + c + d = 10 + 1 + (-14) + 5 = 2
Therefore, the value of a + b + c + d is 2.