Month: February 2024

Week 4 in Precalc 11 – Dividing Radicals

Posted on by averyh2021

This week in Precalc 11 I learnt how to divide radicals. I chose this topic because I found this part of unit 2 to be the most challenging and by writing it out I am helping myself remember these methods.

some things to remember

To divide radicals, the index must be the same in each radical.

  • Divide numerical coefficients by numerical coefficients
  • Divide radicand by radicand
  • Simplify into mixed radical form if possible

 

some examples :

hope this helps 🙂

Week 3 in Precalc 11 – Operations on radicals

Posted on by averyh2021

This week in PreCalc….

To multiply radicals, the index must be the same in each radical.

  • Multiply numerical coefficients by numerical coefficients
  • Multiply radicand by radicand
  • simplify into mixed radical form if possible

it is usually easier to convert each radical to its simplest mixed form before multiplying

Addition and Subtraction:

    • To add or subtract radicals, the radicals must have the same index (root) and the same radicand (expression under the root).
    • Add or subtract the coefficients (numbers outside the radical) while keeping the radicand unchanged.

Multiplication:

    • To multiply radicals, multiply the coefficients together and multiply the radicands together.
    • If the radicals have the same index, you can combine them under one radical.

Division:

    • To divide radicals, divide the coefficients and divide the radicands.
    • Rationalize the denominator if necessary by multiplying the numerator and denominator by the conjugate of the denominator.

Simplification:

    • Simplify radicals by finding perfect square factors in the radicand and taking them out of the radical.
    • Combine like terms under the radical if possible.
    • If there are no more perfect square factors, the radical is simplified.

 

Here are a few examples to illustrate these operations:

     

 

 

.   

I chose this topic because it’s very important that these concepts make sense to me. this week my post is lighter on words and more heave on examples, and this is because I am finding with this unit that I need to see it visually for it to make sense for me.

 

hope this helps 🙂

 

 

 

 

Week 2 in Precalc 11 – Rational Exponents

Posted on by averyh2021

this week in Pre Calc 11, we have been finishing up with our exponents and radical lessons. The biggest issue I have is anything with fractions, which is really bad, so these are notes to myself so that I don’t forget how to do fractions with positive exponents, and radicals. I chose this because this is what I have the biggest problem with and I wanna make sure it stays engraved in my brain.

To start off this was always remember the flower power rule: the root is on the bottom.

When doing anything, multiplying, dividing, adding subtracting. Always remember the Exponent laws that we learnt in grade 10.

  • Zero Exponent Law: a0 = 1.
  • Identity Exponent Law: a1 = a.
  • Product Law: am × an = a. m+n
  • Quotient Law: am/an = a. m-n
  • Negative Exponents Law: a–m = 1/a. m
  • Power of a Power: (am)n = a. mn
  • Power of a Product: (ab)m = amb. m
  • Power of a Quotient: (a/b)m = am/b. m

Don’t get freaked out when things are double square rooted, because they can always be broken down into exponent fractions.

This is what I learnt ;

Rational exponents are exponents that are expressed as fractions. They are a way of representing roots and powers simultaneously. To understand rational exponents, it’s helpful to first understand integer exponents.

Integer exponents are used to indicate repeated multiplication. For example, in a to the power of n, is the base, and is the exponent. If is a positive integer, it indicates that the base should be multiplied by itself times. For instance, a³ means a × a × a

However, what if the exponent is a fraction or a decimal? Rational exponents provide a solution. Rational exponents allow you to represent both roots and powers. Here’s how:

  1. Roots: When the exponent is a fraction in the form 1/m, it represents the (m)th root of the base. For example,  represents the square root of , represents the cube root of , and so on.
  2. powers: When the exponent is any fraction p/q, it can represent both a power and a root. Specifically, ap/q can be thought of as the qth root of a raised to the power of p. For instance, ameans the square root of cubed, while a means the cube root of a squared.

Here are a couple of examples to illustrate:

represents the square root of , which is 2.

 represents the cube root of squared, which is .

Rational exponents provide a convenient notation for expressing powers and roots in a compact form, especially when dealing with non-integer powers and roots.

that’s all

hope this helps 🙂

 

 

 

 

Week 1 in PreCalc 11 – Classifying Real Numbers

Posted on by averyh2021

This week in PreCalc 11 we started re-learning how to classify Real Numbers. I remember learning about in on grade 10 math but I didn’t  remember anything so this was the perfect refresher and I know I won’t forget again.

I chose this topic because I remember being  lost and confused last year when learning about the real number system. I also felt like I was behind in class because I didn’t remember what my peers could remember. by writing out all the rules I am helping future Avery remember all this important information, along with give her something to look back on when she’s confused.

This week I learnt:

Decimal numbers which repeat or terminate can be converted into fractions and are called rational numbers since they can be written as the ratio of two integers

Decimal numbers which are both non-repeating, and non-terminating cannot be converted into fractions and are called irrational numbers

This set of all rational numbers, and the set of all irrational numbers when combined form the set of real numbers these numbers can be represented on a number line

Rational Numbers

Q

  • decimal
  • either terminates or repeats
  • Fraction (ratio of integer)
  • No zero for the denominator

 

Irrational numbers

the character for this is a Q with a line over it.

  • cannot be a fraction
  • must be a decimal
  • never ends
  • non repeating and non terminal decimals

exp.

5.6318…

( if there is no … or relating sign it is not irrational)

  • positive

 

Integer

I or Z

  • has No decimal
  • includes positive numbers , negative numbers, whole numbers , and Zero

 

Whole number

W

0, 1, 2, 3, 4, 5, …

  • No decimals

 

Natural

N

1, 2, 3, 4, 5, 5, …

  • is considered to be  counting numbers (what your parents teach you)
  • No Zero

 

Real numbers

R

Real numbers are both rational and irrational numbers

 

this is a digital version of the chart we drew in class.

 

another good mention is: square roots

specifically to help myself is to remember all positive numbers have two square roots, a positive number, and the other a negative number the positive square root is called the principal square root, and is denoted by the symbol√ .

  • The square root of a perfect square are rational numbers

eg. the square roots of 16 aret be 4 and -4

Note: √16 = 4 only

  • The square roots of a non-perfect square are irrational

eg. the square roots of 17 are √17 and √-17

1²=1

2²=4

3²=9

4²=16

5²=25

6²=36

7²=49

8²=64

9²=81

10²=100

11²=120

12²=144

13²=169

14²=196

15²=225

 

that’s all…

hope that helps 🙂