This week in math class, I learned how to multiply binomials using the double distributive property. A binomial is an expression with two terms, such as (x+1) or (2x−4). To multiply two binomials, we distribute each term in the first binomial to each term in the second binomial, then combine like terms to arrive at the final answer.

Example 1

Problem: Multiply $(x+1)(x+3)$.

Steps:

1. Multiply $x$ by $(x+3)$:

   x(x+3)=x^2+3x

2. Multiply $1$ by (x+3):

   1(x+3)=x+3

3. Combine all the terms:

   x^2+3x+x+3=x^2+4x+3

so,  (x+1)(x+3) = $+3$.

Example 2

Problem: Multiply $(2x+5)(x-2)$.

Steps:

1. Multiply $2x$ by (x−2):

   2x(x−2)=2x^2−4x

2. Multiply $5$ by $(x-2)$:

   5(x−2)=5x−10

3. Combine all the terms:

   2x^2−4x+5x−10=2x^2+x−10

So, (2x+5)(x−2) = 2x^2+x−10

Multiplying binomials using the double distributive property involves distributing each term and combining like terms, which effectively simplifies expressions.

Mrs. Burton taught the class about “Exponent Laws,” and I learned three important ones: multiplication law, division law, and power law.

Multiplication Law:

When multiplying numbers with the same base, you add the exponents together. The base stays the same. For example:

3^2×3^4=3^2+4=3^6

If the bases are different, you multiply the bases normally, and the exponents stay as they are. For example:

2^3×5^3=(2×5)^3=10^3

Division Law:

When dividing numbers with the same base, you subtract the exponents. The base remains the same. For example:

7^5/7^2​=75^−2=7^3

If the bases are different, you divide them and keep the exponents. For example:

8^4/2^4​=(8/2​)^4=4^4

Power Law:

The power law applies when there’s a base with an exponent inside a bracket and another exponent outside the bracket. To solve this, multiply the exponents. For example:

(5^3)^2=5^3×2=5^6

Another example:

(2^4)^3=2^4×3=2^12

I learned how to solve for a missing angle in a right triangle using trigonometry, specifically with the help of “SOH-CAH-TOA.” Mrs. Burton taught me.

Step 1: Label the sides of your triangle.

First, identify and label the sides of your right triangle. The longest side opposite the right angle is the hypotenuse (H). The side across from the angle you’re focusing on is the opposite (O), and the one next to the angle is the adjacent (A). This step ensures you’re set up to use the correct ratios.

Step 2: Choose the right trigonometric ratio.

Next, you need to decide which trigonometric function to use by applying SOH-CAH-TOA:

• SOH: Sine = Opposite / Hypotenuse
• CAH: Cosine = Adjacent / Hypotenuse
• TOA: Tangent = Opposite / Adjacent

Look at the two sides you’ve labeled. For example, if you have measurements for the opposite and adjacent sides, you’ll use the tangent ratio (TOA).

Step 3: Set up the equation.

Now that you’ve chosen the correct ratio, write down the corresponding formula. If you’re using tangent (TOA), for example, the equation would look like:

tan⁡(θ)=O/A​

Here, substitute the actual lengths for O and $A$.

Step 4: Solve for the angle.

To find the angle θ, apply the inverse of the trigonometric function. For tangent, this would be:

θ=tan⁡−1(O/A)

Use the inverse button on your calculator (often labeled as tan−1 or “2nd”) to solve for the angle.

Step 5: Calculate the angle.

Finally, input the values into your scientific calculator. The result will likely be a decimal-round it to the nearest whole number, and indicate rounding by placing a small dot above the equal sign if needed.

And that’s how Mrs. Burton’s helped me find a missing angle using trigonometric ratios.

I got learnt about Trigonometry, including the concept of “SOH-CAH-TOA” and how it helps in solving right angled triangles.

Step 1: Label the sides of the triangle.

The first step is to identify and label the sides of the right triangle. The longest side, across from the right angle, is called the hypotenuse (H). The side directly across from the angle you’re working with is the opposite side (O), and the side next to that angle is the adjacent side (A).

Step 2: Pick the correct trigonometric function.

To decide which function to use, remember the acronym SOH-CAH-TOA:

• SOH stands for Sine, which is the ratio of the Opposite side over the Hypotenuse.
• CAH stands for Cosine, which is the ratio of the Adjacent side over the Hypotenuse.
• TOA stands for Tangent, which is the ratio of the Opposite side over the Adjacent side.

Look at the information you have, like side lengths or angles, and choose the function that fits the sides involved.

Step 3: Write out the equation.

Let’s say you’re using the sine function and the angle is 35 degrees. The equation would look like this:

sin⁡(35∘)=x/10m​

Here, $x$ is the unknown side you want to solve for, and 10 meters is the length of the hypotenuse.

Step 4: Solve for the variable.

To isolate the variable, multiply both sides by the length of the hypotenuse. In this case, it would be:

10×sin⁡(35∘)=x

Step 5: Calculate the result.

Enter this equation into your calculator. For instance:

10×sin⁡(35∘)=5.7m

This gives you the length of the missing side. And that’s how you can use trigonometric ratios to find unknown sides of a right-angled triangle!

Scientific Notation – Large numbers example:

Let’s say we have a number like 9,300,000. In Standard Notation, that’s the number we would write out in full, but in Scientific Notation, it becomes 9.3 x 10⁶. Here’s how it works: you place the decimal between the 9 and the 3, because the first part (the “coefficient”) has to be a number between 1 and 10. After that, the exponent (6) tells us how many places the decimal moved from its original position to get to this format. So in this case, the decimal moves 6 places to the left to get from 9,300,000 to 9.3.

Scientific Notation – Small numbers example:

For smaller numbers, it’s a similar process, but we move in the opposite direction. For example, if we have 0.00047, the Scientific Notation would be 4.7 x 10⁻⁴. We moved the decimal 4 places to the right this time, and because we moved to the right, the exponent is negative. Just like before, the coefficient (4.7) has to be a number between 1 and 10. So, 4.7 x 10⁻⁴ is a more compact way to write 0.00047.