Evaluate the expression C = 8x^3 + 12x^3 + 6x + 1 at x = 1/2.

Step-by-step solution

1. 1

   Combine like terms first. $8x^3$ and $12x^3$ have the same variable part, so they add directly:

   $8x^3+12x^3=20x^3$

   Doing this before substituting means computing $\left(\tfrac12\right)^3$ once instead of twice.

2. 2

   Write the reduced expression.

   $C=20x^3+6x+1$

3. 3

   Compute the cube of the fraction.

   $\left(\frac12\right)^3=\frac{1}{8}\quad\Longrightarrow\quad 20\cdot\frac18=\frac{20}{8}=\frac52$

   Remember the exponent applies to numerator and denominator alike: $\tfrac{1^3}{2^3}=\tfrac18$.

4. 4

   Compute the linear term.

   $6\cdot\frac12=3$

5. 5

   Add the three pieces over a common denominator.

   $C=\frac52+3+1=\frac52+4=\frac52+\frac82=\frac{13}{2}$

6. 6

   Check as a decimal. $20(0.125)+6(0.5)+1=2.5+3+1=6.5=\tfrac{13}{2}$. $\checkmark$