Evaluate 1/3 + square root(3)/12 + 1/16 + square root (3)/64.
I hope this makes sense. Right now we're learning sequences and series and I'm really confused on this question.
1/3 + √3/12 + 1/16 + √3/64
Using a common denominator of 64*3, we have
(64 + 16√3 + 12 + 3√3)/192
(76+19√3)/192
or
19(4+√3)/192
To evaluate the given expression 1/3 + square root(3)/12 + 1/16 + square root (3)/64, we need to find a common denominator and then combine the terms.
Step 1: Find the common denominator.
The denominators in the expression are 3, 12, 16, and 64. The least common multiple (LCM) of these numbers is 96. So, we will convert all the fractions to have a denominator of 96.
Step 2: Convert the fractions to have a common denominator.
1/3 can be converted to have a denominator of 96 by multiplying the numerator and denominator by 32. This gives us (32/32) * 1/3 = 32/96.
Square root(3)/12 can be converted to have a denominator of 96 by multiplying the numerator and denominator by 8. This gives us (8/8) * square root(3)/12 = 8 * square root(3)/96.
1/16 can be converted to have a denominator of 96 by multiplying the numerator and denominator by 6. This gives us (6/6) * 1/16 = 6/96.
Square root(3)/64 can be converted to have a denominator of 96 by multiplying the numerator and denominator by 3. This gives us (3/3) * square root(3)/64 = 3 * square root(3)/192.
Step 3: Combine the fractions.
Now, we can add the fractions with the common denominator:
32/96 + 8 * square root(3)/96 + 6/96 + 3 * square root(3)/192
To add the fractions, we add the numerators and keep the common denominator:
(32 + 8 * square root(3) + 6 + 3 * square root(3))/96
Simplifying the numerator:
38 + 11 * square root(3)
So, the simplified answer is (38 + 11 * square root(3))/96.