Leah here from Leah4sci.com/MCAT and in this video, I'll show you how to solve Squares
and Square Roots on the MCAT without a calculator. When it comes to simple Math like Multiplication
and Division, it's easy to find shortcuts but it's also easy to fall back on the long
form Multiplication and the long Division. When it comes to Squares and Square Roots,
it's even more important to know and understand the shortcuts because you'll have nothing
to fall back on when you're not allowed to use your calculator.
One of my MCAT students brought up this question in a recent tutoring session.
Find the hydronium ion concentration in 0.05 molar solution of acetic acid given an acid
dissociation constant of one point eight times ten to the minus five (1.8x10^-5).
I will discuss this concept in my MCAT Chemistry videos at http://leah4sci.com/MCATChemistry,
but for now, let's focus on the Math. We have a zero point zero five (0.05) molar solution
and a ka of one point eight times ten to the minus five (1.8x10^-5). We'll set up the equation
using the Chemistry trick of ka is equal to x squared divided by the Initial Molarity
(ka=X²/Mi) with H+ concentration is equal to X. I like to simplify my equations by isolating
X and then plugging in the numbers to solve. So we'll multiply both sides by the Initial
Molarity (Mi) and then take the Square Root of each side to isolate X. This gives me an
equation as follows; X is equal to the Square Root of Initial Molarity which is point o'five
(.05) and a ka of one point eight times ten to the minus five, and this is where my students
froze. We have a square root and we have scientific notation so how do you proceed?
I'm assuming that you've already watched my video on how to multiply scientific notation,
if you haven't you can find it on my website at http://leah4sci.com/MCATMATH.
Starting with Multiplication, we have one point five times one point eight times ten
to the minus five (0.05 x 1.8 x 10^-5), this can appear daunting but don't let it scare
you. Remember the trick of times 10 divided by 10 that keeps the overall value the same?
That's exactly how we are going to simplify this. And here's how we're gonna do it.
To get rid of the decimal from 0.05, we have to move the decimal two spaces to the right
which means we are multiplying by a hundred (100). To make up for it and keep the overall
value the same, we do the same thing to the number on the left but in reverse! We'll DIVIDE
by 100; the way we divide by factors of 10 is dropping the exponent 1 value for every
factor of 10. So if I have 10 to the minus of 5 and I wanna make that smaller by a factor
of a hundred, 10 to the minus 5 becomes 10 to the minus 6 (10^-6), for ten and ten to
the minus seven (10^-7) for one hundred (100) because the difference here is one times ten
to the second (1 x 10^2) which is a hundred. This gives me a new number to work with, five
times one point eight times ten to the minus seven (5 x 1.8 x 10^-7), much easier to deal
with but we can simplify this further. 1.8 is approximately 2 giving me a new value to
calculate five times two times ten to the minus seven (5 x 2 x 10^-7). Well 5x2 is 10,
the scientific notation we want 1 number and a decimal. So once again we'll do times 10
divided by 10. Ten divided by ten gives me one (10/10=1), Ten to the minus seven times
ten, we move one decimal to the right or raised the power by one so negative seven goes up
to negative six giving me an answer of one times ten to the minus six (1x10^-6).
Now going back to the original question, we have x is equal to 1 times ten to the minus
6. We wanna take the square root of the number and the square root of the power and put it
together. The square root of one is one so that's easy. The square root of the power
means you simply divide it by two if it's an even number. The square root of 10 to the
minus six is simply six divided by two is three which means one times ten to the minus
three (1x10^-3) and that is equal to our Hydronium or H3O+ concentration.
Another way that you can do this is to think about what you know of the Square Root. If
you are given the square root of X, that's the same thing as raising X to the half power
(x^1/2). If we apply this to our number we have one times ten to the minus six raising
it to the half power  (1x10^-6) ^1/2. When your exponent is raised to another exponent,
all you do is multiply the exponents. So here we have negative six as our first exponent
times one over two which is our second exponent (-6x1/2). Negative six over two is negative
six divided by two and that ones again gives us negative 3 (-3) which is our answer one
times ten to the minus three (1x10^-3). Let's apply this to another example, say you're
told; Find the maximum height a feather can reach
f it takes off the ground with an initial velocity of 9.3 x 10^-4 m/s. Ignore air resistance.
I'll go through the Kinematics portion of this question in my Physics series at leah4sci.com/MCATPhysics
but for now let's focus on the Math. The formula we'll use is V initial squared is equal to
V final squared plus 2a delta x (Vi^2 = Vf^2 + 2a ∆x). Since we're dealing with height
and gravitational acceleration, we'll change the x to a y and positive a becomes negative
g for gravitational acceleration. When an object reaches its Maximum Height the velocity
at the top will be zero so we can cross it out. We're solving for delta y so let's isolate
that by moving everything to the other side. Minus v initial squared gives us a new expression
negative v initial squared is equal to negative 2g delta y. Divide by negative 2g on both
sides and cancel out the negative for a final expression of delta y is equal v initial squared
over 2g  (∆Y=Vi^2/2g). Now let's plug in the numbers. V initial is nine point three
time ten to the minus four (9.3 x 10^-4) so we have nine point three times ten to the
minus four squared (9x10^-4)^2, divide that by 2g, the gravitational acceleration is nine
point eight meters per second squared (9.8m/s^2) but on the MCAT you're fine using 10. We'll
skip the units for the sake of focusing just on Math. You have two options here; the denominator
is easy, two times ten is twenty but for the numerator we have two options. Option number
1: We can take that square and distribute it; Option number 2: We can actually write
it up as the number times itself. The concept or the ultimate Math to do will be the same
so let's show how to do it by writing it out. We have Nine point three times ten to the
minus four times Nine point three times ten to the minus four. Now remember, on the MCAT
you are allowed to round so we'll round Nine point three down to Nine and then look for
an answer that is slightly greater to make up for rounding them. When multiplying the
exponents, you multiply the individual numbers so here we have nine times nine which is Eighty
One and then we add the exponents. Since both are ten to the minus four we simply add negative
four plus negative four giving me negative eight and that will be times ten to the minus
eight. Another option would have been to just distribute
the square as follows; if we square nine point three, we have nine times nine which is eighty
one, for ten to the minus four square, since we're taking the power and raising it to a
power, we multiply the numbers. Negative four times two is negative eight. So once again
we get a simplified version of this number that is Eighty One times Ten to the minus
Eight (81 x 10^-8). So let's continue, our V initial squared is
simplified to Eighty One times Ten to the minus Eight (81 x 10^-8) and we're dividing
that by 20. Assuming that Eighty One is approximately eighty, remember we're simplifying, when you
have two numbers that end in Zero, whatever you do to the top, you can do to the bottom
and keep your number constant. So if I have Eighty over Ten giving me Eight and Twenty
over ten giving me Two then I've simplified it to Eight over Two. Another way to look
at this is you can simply cancel the Zeros. Eight over two is Four so that'll give me
Four times Ten to the minus Eight (4 x 10^-8). This was a very simplified way of doing the
question but look what happened! I punched the initial numbers into the calculator and
got an answer of 4.3245x10^-8. Now let me ask you, was it worth saving time to get an
answer that is so close even for an MCAT question? Totally!
Be sure to join me on the next video where I break down MCAT Fractions, Percentages and
Ratios. Are you stuck on a specific MCAT topic? I
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