Below is the transcript of my tutorial video MCAT Math video 4 – Scientific Notation for Addition, Subtraction, Multiplication and Division.
Leah here from Leah4sci.com/MCAT and in this video, I’ll show you how to use Scientific Notation in MCAT style questions without a calculator when it comes to addition, subtraction, multiplication and division.
When it comes to adding and subtracting exponents as long as you’re dealing with the same power all you have to do is add and subtract the coefficient and keep the power. So for example, say you’re given 4.7×10¯³ + 2.3×10¯³ . Both numbers in the question are times 10 to the minus 3 so all we have to do is add 4.7 and 2.3.
Let’s We’ll start with the coefficient, 4 and 2 is 6, point 7 and point 3 is 1 gives us a total of 7.0×10¯³. Ten to the minus 3 hasn’t changed and all I did was add the two numbers.
When the powers are not the same but they’re close enough, you want to turn them into the same power and then treat it the same way. Let’s keep it simple, say I have 5×10¯³ + 6×10¯⁴ . I can’t simply add 5 and 6 because I’m dealing with different powers.
So what I wanna do is turn the first number into the power of negative 4 or the second number into a power of negative 3. In an earlier video I showed you a trick for multiplying and dividing by a factor of 10 by simply moving the decimal. You can find that video along with my entire MCAT Math without a Calculator series on my website leah4sci.com/MCATMATH.
Scientific Notation is simply another way to represent powers of 10 and if I treat it that way, the trick of moving decimals still applies. If I have the number like 10¯³ the way to make this to 10¯⁴ is to divide it by 10. Now remember the trick I showed you, if I have an example and I do times 10 divided by 10, I don’t change the identity of the example.
So if I take 10¯³ and divide that by 10, I get 10¯⁴. I have to justify that by multiplying this number (5) by 10. 5×10 is 50. Changing 5×10¯³ to be 50×10¯⁴ has the same exact value. Now that I have 50×10¯⁴ I can add that to 6×10¯⁴ because they have the same power. 50 + 6 is 56 and we still keep 10¯⁴.
Now in typical Scientific Notation, you want just one number immediately followed by a decimal. So I have to do that times ten divided by ten again or simply think of it as when I decrease the number, meaning I move the tens place decimal 1 space to the left, I increase my exponent value. So 56 divided by 10 is 5.6, 10¯⁴ times 10 is 10¯³. In other words, I use that trick again of times 10 divided by 10 and quickly got my answer. So the final answer for this question is 5.6×10¯³.
Multiplication of Exponents may show up on your MCAT in question like this:
Find the NO3- concentration when 6.8×10¯⁷ moles NANO₃ are added to 5.1×10¯³ L of H20.
Finding molarities on topic that will be discussed in the Chemistry series at leah4sci.com/MCATChemistry, but for now let’s focus on Math. We want to solve for Molarity which is mol/L (M=mol/L). On the numbers out of the the question, we have an equation like this :
M=mol/L = 6.8×10¯⁷ / 5.1×10¯³ mol/L
This throws off so many students because they make it harder than it has to. So let’s talk about the rules then let’s solve the problem. When it comes to Multiplication, all you do is multiply the number and then ADD the exponents.
(2×10⁴)(4×10²)=8×10⁶ –> In this case we have 2×4 which is 8. For the exponents 10⁴ and 10² we add 4 and 2 which is 6 giving us an answer of 8×10⁶.
For division, you divide your numbers and then subtract your exponents. In this case we have:
3 x 10⁶ ÷ 6 x 10³ , we’ll set that up as follows: 3 divided by 6 that’s 3/6, we can simplify that to 1/3 which is 0.33, this is the fraction that you have to know. If you don’t know it yet, grab my MCAT Math guide on my website https://leah4sci.com/MCATMath.
Next we subtract the exponents, we have 10⁶ and 10³, 6 minus 3 is 3 giving me an answer of .5 x 10³. Now this is not proper Scientific Notation so we’ll actually move the decimal 1 space to the right multiplying by 10 and decrease the exponent by 1 number which is essentially dividing by 10, in other words we’re multiplying and dividing by 10 giving me an actual answer of .5×10² or simply 500.
Now back to our question. For a question like this, the MCAT is not looking for the exact value, instead they are looking for something close enough and you’ll typically see answers that have the powers all over the place.
So we’re looking for an approximate number but the correct power. The power in this case is 10¯⁷ divided by 10¯³ so we have 7 minus negative 3. Negative 7 minus negative 3, Minus Negative cancels out so it’s really Negative 7 plus 3 (-7+3) giving me a new exponent of 10¯⁴.
The numbers themselves 6.8 is approximately 7, 5.1 is approximately 5 so we have 7/5 x 10¯⁴. This is not an appropriate answer and in the next video I’ll break down fractions and ratios in more detail but for now look at it this way; 7/5 is really 5/5 + 2/5 5/5 is 1 so we have 1 and 2/5 x 10¯⁴. Two over Five (2/5) is a fraction you should recognize because one over five (1/5) is point two (.2), therefore Two over Five (2/5) is point four (.4).
This gives us a final answer of 1.4 x 10¯⁴. Punching the initial numbers into the calculator, I got 1.33 x 10¯⁴ which on the MCAT is close enough.
Be sure to join me in the next video where I show you how to tackle Squares and square roots when it comes to the MCAT.
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Watch MCAT Math video 4 – Scientific Notation for Addition, Subtraction, Multiplication and Division.