A base that is raised to a negative exponent has an integral exponent. 

The general rule for when a base is raised to a negative exponent This is because exponents have a pattern of being divisible by the base, so if continued past exponent 0, you begin to get fractions

You flip it from the numerator to the denominator or the denominator to the numerator. This flip makes the exponent go from a negative exponent to a positive exponent. 

This law remains the same for evaluating different expression

When there are numbers with positive exponents and negative exponents, only the bases with the negative exponents need to move.

The general rule of fraction base to a negative exponent is

This is because 

Evaluating an expression with a fraction for a base and a negative exponent 

You flip the fraction to make the outer negative exponent, a positive exponent. Then proceed to continue simplifying

This unit was very challenging, it called for many different exponent laws for different situations. Once it is explained, though, and I figured out when to use which exponent law, I got the hang of it. When many different of these exponent laws are seen in the same expression, it may seem over whelming to simplify or evaluate, but if you follow through with each law, it become quite easy.

Radical 

A radical is an expression in the form ofAn example of a radical is It includes a index, radicand, and radical symbol, making all together a radical. In this case the index is 5 and the radicand is 125. 

Note:

If the index is not written, as in square root, it is assumed to be 2.

The index is the number of times the radical must be multiplied by itself to equal the radicand.

An entire radical is when the number is entirely under the root symbol or the radical sign. Whereas a mixed radical isn’t. A mixed radical is a form of simplifying an entire radical.

 

 

 

Converting an entire radical to a mixed radical

To convert or simplify an entire radical, you need to find two factors of the number, one being a perfect root. For example,

4 being the perfect root in this case. Then each of those factors become a radical, leaving you with radical that isn’t a whole number (because it is not perfect, and becomes a decimal if rooted). You are then left with a mixed radical. 

If you wish to simplify it futher, you need to repeat the process by finding another perfect root that is a factor of the radicand. 

Converting a mixed radical back to an entire radical

You need to put the number on the outside of the radical sing back under the sign. To do this, you use the index as an exponent for that number. Than multiply it by the radicand. You end up with an entire radical.

 

This week we learned about Radicals, mixed radicals to entire radicals, and entire radicals to mixed, no matter what the index is. One challenge with this lesson is remembering all the steps. In order to make sure that your end result is correct, you need to make sure you don’t forget any step and write everything down so you do not get confused. If you forget to write out each factor as a radical, you can forget to find the roots, therefore leaving you confused. Once I got the hang of it, it became a very easy lesson/subject.

GCF

GCF or Greatest Common Factor is the highest number that can divide exactly into a set of numbers. (For example: 7 is the GCF of 14 and 49, 7 divide into 14 twice and divides into 49 seven times.)

Some numbers are easy enough to do the math mentally (For example: 3 and 9, you can quickly see that the highest factor that goes into both those numbers is 3), but when there are higher numbers or more than a set of two numbers, you can use prime factorization in order to help you find the GCF.

The First Step

Find the prime factorization of the given numbers. (Prime Factorization – Prime factors of a number, the sum is equal to the number)

The Second Step

Find the prime factors in common, also choose the smallest exponent between the common prime factors. 

The Last Step

Multiply all the common factors and the smallest exponents in order to find the GCF. 

(The same steps no matter how high the numbers or how many numbers you are working with.)

LCM

LCM or Lowest Common Multiple is the lowest number that is a multiple of a set of numbers. (For Example: 12 is the lowest common multiple of 2, 3, and 4.)

Again, for finding LCM, you could probably do the math mentally when the numbers are small like 2, 3, and 4. But for higher numbers, instead of listing out all the multiples of each number and finding which is the lowest multiple in common, we can use prime factorization.

The First Step

Find the prime factors of each number.

The Last Step

Take all the prime factors and their highest exponent. Multiply to find the LCM.

This week, we learned about prime factors, prime factorization, and how to apply prime factorization when trying to find the GCF and LCM. One challenge with this lesson is that there is a lot of mental math and math without a calculator. You need to figure out a lot when it comes to factors especially, like doing the math on whether a number divides exactly into another number when trying to find the prime factors. It is a very good thing to learn because it trains our minds and helps us learn how to do math without a calculator.