Math 10 Honors Numbers Summary

Over the course of this unit, we have learned prime factors, GCFs and LCMs, the real number system and its branches, and radicals of all sorts.

During the numbers unit, we learned how to prime factorize small and large numbers, finding the GCF (Greatest Common Factor) of multiple numbers, as finding the LCM (Least Common Multiple) of the same process. When prime factorizing, we either use a factor tree or division table to divide a number by its prime factors. Similarly, when we have the prime factors of certain numbers, we can either multiply common factors or find the greatest factor to find the LCM and GCF.

In the real number system, we learned the two types of real numbers: rational and irrational. From there, there were three smaller sets inside the rational numbers: the integers, the whole numbers, and the natural numbers. I had known this from grade 8 and personal tutoring, but I did not know that there was a set of imaginary numbers such as i.

In the radicals unit, we extended our comprehension of square roots to cube roots and indexes beyond. We first the vocabulary of each part of a radical, such as the radicand (the big number inside the radical) and the index (small number outside the radical). We learned that the square root or any other even index (root) can have a positive and negative answer. We also learned that a positive radicand can only have a positive cube or odd root, such as \sqrt[3] {8} only being 2. Likewise, a negative radicand can only have a negative odd root, such as \sqrt[3] {-8} only being -2.

Furthermore, we learned how to convert mixed radicals into entire radicals and vice versa. Mixed radicals are radicals where there is a coefficient in front of the radical itself. When converting entire radicals into mixed radicals, you have to find a factor that is a perfect square, cube, etc. depending on the index indicated. For example, when you have an entire radical like \sqrt {72}, you have to find a perfect square factor in 72, which is 36. The other factor would be 2, so now we can say that \sqrt {72}=\sqrt {36} \cdot \sqrt {2}. The square root of 36 is 6, and the square root of 2 is irrational. So now we write our equation like this: 6 \sqrt {2}.

Much like above, when converting mixed radicals such as 2\sqrt {2} to entire radicals, we follow these certain steps. First, we want to convert the coefficient back to a radical, so we convert it to a perfect radical. This makes it \sqrt {4} \cdot \sqrt {2} or \sqrt {4 \cdot 2}. Now we multiply the radicands, which makes the entire radical \sqrt {8}.

Last but not least, we learned when the radicand is a variable. Variables differ from numbers, where we have to find perfect exponents. For variables, we only have to find multiples of the index. For example, when doing \sqrt {a^7}, we split like this: \sqrt {a^6} \cdot \sqrt {a}. We do this because 6 is the biggest multiple of 2 in the number 7. Now, the mixed radical becomes a^6 \sqrt {a}.

This is the brief summary of unit #1.

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