Adding Radicals

Adding radicals in math is actually a lot more simple than I originally thought. Radicals (or roots) are expressions that involve the root of a number. The most common type is the square root, but there are also cube roots, fourth roots, and so on. Here’s how I approach adding radicals:

Step 1: Simplify the Radicals

The first thing I do is simplify the radicals. Simplifying a radical means breaking it down into its simplest form. For example, I take √50 and simplify it to 5√2 because you can pull out a five. This is because when factored, 50 breaks down into the prime factors: 5, 5, 2. and as it’s a square root, you need two of the same number to pull them out.

Step 2: Ensure Like Radicals

Next, I check to make sure the radicals are the same, which means they have the same radicand (the number under the radical sign) and the same degree of the root (square, cube, fourth, etc). For instance, 3√2 and 5√2 are like radicals because they both have the radicand 2 and are both square roots. On the other hand, √2 and √3 are not like radicals and can’t be added directly.

Step 3: Add the Coefficients

Once I have like radicals, I add them by simply adding their coefficients (the numbers in front of the radicals) while keeping the radical part the same. For example: 3√2 + 5√2 = (3 + 5)√2 = 8√2. Overall pretty simple, but also very important to remember.

Example Problem

1. Example 1: Simple Addition2√3 + 4√3​
   Since the radicals are the same, I just add the coefficients:2 + 4 = 6So,

   2√3 + 4√3​ = 6√3

Sine Law:

General Idea

The sine law is a rule that helps you find missing sides or angles in a triangle. It says that if you take the length of a side and divide it by the sine of the angle opposite that side, you’ll get the same number for all three sides of the triangle.

Formula

a/sin⁡A   =   b/sin⁡B   =   c/sin⁡C

—-​

Cosine Law:

General Idea

The Law of Cosines is like the Pythagorean theorem but for all triangles, not just right-angled ones. It helps you find a side or angle when you know the other sides and angles.

Formula

c^2 = a^2 + b^2 − 2abcos⁡C

—-

When to Use Each Law:

• Sine Law: Use this when you know:
  • Two angles and one side
  • Two sides and a non-included angle

• Cosine Law: Use this when you know:
  • All three sides
  • Two sides and the included angle

When adding algebraic fractions…

1. Find a Common Denominator: The first step is to find a common denominator. If you want to save yourself the work later on, you can identify the least common multiple of the denominators of the fractions now. This common denominator will allow you to add the fractions together.
2. Rewrite the Fractions: Rewrite each fraction so that they all have the same denominator. To do this, multiply both the numerator and the denominator of each fraction by the necessary factor(s) to make the denominators the same. You can also write it as all one big fraction with the single denominator, but I prefer to keep them separate at this stage.
3. Add the Numerators: Once the fractions have the same denominator, add their numerators together while keeping the common denominator unchanged. Remember to simplify the resulting expression by combining like terms.
4. Simplify the Fraction (if possible): If the expression in the numerator can be simplified further by factoring or canceling out common factors, then do so.
5. Check for Restrictions: When there are variables involved, you need to check if there are any restrictions on the variables that would make the expression undefined. If so, state the restrictions.
6. Verify (Optional): To be absolutely sure, it can be good to verify by inputting a number into the original expression as well as the simplified one,

This all works similarly with subtraction, though when it comes to algebraic fraction, it’s always safest to turn subtraction expressions into addition with flipped positive/negative signs in the numerator.

Among other things, this week in math we looked at inequalities. The word inequality means quite simply, that the two sides of an equation are not equal. Inequalities are used to compare values. We use the symbol “<” (less than) and “>” (greater than) to show which side is higher in value. This can be transferred to graphs relatively easily. If we know that…

y > 3 we also know that the variable “y” must be a value greater than 3. This would, on a graph, leave the entire space above y = 3 shaded in as that is the infinite space that a point could be placed.  When showing this on a graph, the line at y = 3 has to be dotted to show that the value must be higher than 3 and not 3 itself.

Now, if there’s a line underneath the symbol like “≤” then that simply means “greater than or equal to…” It just indicates that the value can also include the number itself. To demonstrate this on a graph, the line would just become solid instead of broken.

Now, when applied to linear equations, it’s actually quite simple, as our example from before (y = 3) was also linear, just more simple. Take the line y = 2x +3. It’s incredibly easy, just a straight line with a slope of 2/1 and a y-intercept of 3.

Now if we swap the equals sign for an inequality, like y ≤ 2x + 3, then all that means is that the value must be below the line.

* Another thing to always remember is that the inequality is like an arrow: with > pointing to the right on the number line, meaning it is a greater than symbol with < pointing towards the negatives on a number line meaning it is a less than symbol.