Converting Numbers to Scientific and Engineering Notation
Sure, let's solve both parts of the question:
a) Convert 0.000 000 035 to Scientific Notation
Scientific notation is a way to express numbers that are too large or too small to be conveniently written in decimal form. It is typically written in the form of \( m \times 10^n \), where \( m \) is a number greater than or equal to 1 but less than 10, and \( n \) is an integer.
To convert 0.000 000 035 to scientific notation, we find the most significant digit (which is 3 in this case) and count how many places we need to move the decimal point to get to that digit. We move the decimal 8 places to the right, which gives us a factor of \( 10^{-8} \). So, 0.000 000 035 in scientific notation is \( 3.5 \times 10^{-8} \).
b) Calculate \( (3.4 \times 10^9) \times (4.9 \times 10^{11}) \), giving your answer in Engineering Notation
Engineering notation is a variant of scientific notation where the exponent \( n \) is divisible by 3. This is so the values can easily be interpreted in terms of thousands, millions, billions, etc.
Let's multiply the two numbers, keeping in mind that we multiply the coefficients and add the exponents:
\( (3.4 \times 10^9) \times (4.9 \times 10^{11}) \)
\( = 3.4 \times 4.9 \times 10^{9+11} \)
\( = 16.66 \times 10^{20} \)
Now we adjust the coefficient to be between 1 and 1000 by moving the decimal point, which also adjusts the exponent to be a multiple of three:
\( = 16.66 \times 10^{20} \) is already in a form where the exponent is a multiple of 3, but usually, in engineering notation, we prefer to have the coefficient in the 1.000 to 999.999 range. Let's further adjust it to meet this criterion:
\( = 16.66 \times 10^{18} \times 10^{2} \)
\( = 1.666 \times 10^1 \times 10^{18} \)
\( = 1.666 \times 10^{19} \)
Now, \( 1.666 \times 10^{19} \) is in proper engineering notation.
