To express the number [tex]\( 780,000,000 \)[/tex] in scientific notation, we need to write it in the form [tex]\( a \times 10^b \)[/tex], where [tex]\( a \)[/tex] is a number greater than or equal to 1 but less than 10, and [tex]\( b \)[/tex] is an integer.
Here’s a step-by-step guide to accomplish this:
1. Identify the Significant Figures: Place the decimal point after the first non-zero digit. For 780,000,000, the first non-zero digit is 7, followed by 8. Thus, the value will be [tex]\( 7.8 \)[/tex].
2. Count the Exponent: Count how many places the decimal point has been moved from its original position (which was at the end of the number). In [tex]\( 780,000,000 \)[/tex], the decimal point is moved 8 places to the left to place it between 7 and 8. Therefore, the exponent will be 8.
3. Combine to Form the Scientific Notation: The number is now written as [tex]\( 7.8 \times 10^8 \)[/tex].
Now, let’s match it with the given choices:
A) [tex]\(0.78 \times 10^9\)[/tex]
- This is not correct since the coefficient is 0.78 which is not between 1 and 10.
B) [tex]\(0.78 \times 10^{-9}\)[/tex]
- This is not correct since both the coefficient and the exponent are not matching.
C) [tex]\(7.8 \times 10^8\)[/tex]
- This matches our derived scientific notation exactly.
D) [tex]\(7.8 \times 10^{-8}\)[/tex]
- The exponent here is negative and therefore does not match.
E) [tex]\(78 \times 10^7\)[/tex]
- Although this equals the original number [tex]\( 780,000,000 \)[/tex], the coefficient 78 is not in the correct range for scientific notation (1 ≤ a < 10).
Thus, the correct scientific notation for [tex]\( 780,000,000 \)[/tex] is [tex]\( \boxed{7.8 \times 10^8} \)[/tex], which corresponds to option C.
