Category Archives: Math 10

This week in math 10 we learned about right pyramids, cones, and how to calculate slant height.

To calculate the slant height of right pyramids, you need to have the height and half of the length. Then you can use the Pythagorean theorem to find the hypotenuse (your slant height). Then you can use your slant height when you calculate the total surface area or volume.

Example: When you have the height (3 in this case) and the side length (8), you can calculate the slant height. Take half the side length (4) and use it and the height in the Pythagorean theorem. In this case + = 25 = . Therefore 5 = c, and 5 is your slant height. You can then plug the slant height into the surface area and volume equations.

The Pythagorean Theorem principle is also used when calculating the slant height of a cone. The only difference is instead of using half the side length and the height to find the hypotenuse (slant height), you use the radius of the base and the height.

This week in math we learned how to convert units of measurement in a simple way between both longer/shorter units within the same system, and between the metric and imperial systems.

Easy to do, you simply need to know how many of one unit is equal to the other unit.

Example: The units that you wish to convert go on the left (6780cm), the units that you wish to convert to go in the numerator position (1m) and however many units of one equal the other (100cm) in the denominator.

Then simply evaluate your expression

The same can be done for converting units between the metric and imperial systems.

Example: If you know 1ft is equal to 0.3048m, then you repeat the same steps again.

And evaluate.

This week in math 10 we learned about integral exponents (negative exponents) and how to get rid of them.

To get rid of negative exponents you need to take the base and flip it.

Example: Whatever your variable or base is, it on it’s own is equal to itself over 1. You just don’t write it as such. So when you flip it, you take whatever is in the numerator, and move it to the denominator, or the other way around. This switch makes the exponent positive.

This same principle of moving a base with an integral exponent to the other position applies to solving other expressions as well. This includes expressions where there are bases/exponents in both the numerator and denominator.

Example: Using the same principle of moving a base with a negative exponent to the other position, we move our negative exponent from the numerator to the denominator, changing the negative to a positive, and then evaluating as you would a normal expression.

This week in math we learned how to convert entire radicals into mixed radicals.

First, find two factors of your radicand, one being a perfect square. Then you will have two radicals, one giving you a whole number, and the other not. This will give you your mixed radical.

Example: In this case you need to find a factor of 75 that is also a perfect square. 25 x 3 is equal to 75, and 25 is also a perfect square. Therefore you square root 25 and move the whole number to the outside of the radical. Therefore 5 x the square root of 3 is equal to the square root of 75.

The same thing can be done with radicals which contain any index. Instead of finding a factor of the radicand that is a perfect square, you find a factor that is either a perfect cube, perfect 4th, etc. depending on the index.

Example: To simplify the cube root of 48, you first need to find a factor of 48 that is a perfect cube. 8 x 6 is equal to 48, and 8 is a perfect cube. You then cube root 8, and again move the number to the outside of the radical. So 2 x the cube root of 6 is equal to the cube root of 48.

This week in math 10 we learned how to find the GFC and LCM of two or more numbers.

GFC (Greatest Common Multiple):

First, find the prime factorization of your numbers. Then take all of your common prime factors and multiply them together to find your GCF. If common factors have exponents, take the lesser of the two exponents when you go to multiply the common factors together.

Example: Take your two numbers (48 and 72 here) and find the prime factorization.

(prime factorization)

Next, you take the prime factors (circled above) and find common factors. Take whichever common factors have the LOWEST exponent, and multiply the result together like so:

Therefore the GCF of 48 and 72 is 24.

LCM (Lowest Common Multiple):

To begin, once again find the prime factorization of your numbers. Find all of the prime factors as you did with the GCF, however instead of only taking common factors, you take ALL the prime factors. If two or more prime factors are the same, you take the one with the GREATEST exponent this time. Then, simply multiply all the factors together to get the LCM.

Example: Take your numbers (18 and 63 here) and find the prime factorization.

(prime factorization)

Next, take all prime factors and multiply them. Keeping in mind if two factors are the same, you are taking the one with the lowest exponent.

(Both 18 and 63 had a prime factor of 3, but instead of taking both, only one was taken. Exponents were the same so were not a factor in the decision.)

Therefore the LCM of 18 and 63 is 126.