Explanation

If θ = sin⁻¹x, then sin(θ) = x.

From the identity:

cos²θ + sin²θ = 1,

⇒ cos(θ) = √(1 − x²).

Explanation

Step 1: Determine the value of each power of j

j¹⁰ = (j²)⁵ = (-1)⁵ = -1.

j¹⁰⁰ = (j²)⁵⁰ = (-1)⁵⁰ = 1.

j¹⁰⁰⁰ = (j²)⁵⁰⁰ = (-1)⁵⁰⁰ = 1.

Step 2: Calculate the expression

1 + j¹⁰ + j¹⁰⁰ - j¹⁰⁰⁰ = 1 + (-1) + 1 - 1 = 0.

Explanation

j¹⁰⁵ = j^(4*26 + 1) 

= (j⁴)^26 * j¹ 

= 1²⁶ * j 

= j.

Explanation

"3x" usually represents 3 rectangles (each representing x).

"+2" typically means 2 small unit squares (each representing 1).

So, a model showing 3 long bars and 2 small squares matches 3x + 2.

Explanation

Let's solve the expression 3(a ÷ b)^c with the given values: a = 36, b = 12, and c = 2.

First, calculate a ÷ b = 36 ÷ 12 = 3.

Then, raise it to the power of c: 3^2 = 9.

Finally, multiply by 3: 3 * 9 = 27.

Explanation

Given the equation x^2 + y^2 = 1, the range of y values is [-1, 1] since y ranges from -1 to 1 as x varies.

Explanation

Step 1: Recall the formula for torque

Torque (τ) is related to the change in angular momentum (L) over time (t) by the formula: τ = ΔL / Δt

Step 2: Calculate the change in angular momentum

ΔL = 5L - 2L = 3L

Step 3: Calculate the torque

τ = ΔL / Δt = 3L / 3 = L

Explanation

In an arithmetic progression, the sum of n terms between a and b is given by: Sum = n(a+b)/2

This formula uses the first and last terms to find the total of evenly spaced values.

Explanation

To find the sum, we need to calculate each term:

i^101 = i (since i^1 = i and the powers repeat every 4 terms)

i^102 = i^2 = -1

i^103 = i^3 = -i

i^104 = i^4 = 1

Now, let's add them up:

i + (-1) + (-i) + 1 = 0

Explanation

The square root of any prime number is not a perfect square.

Therefore, √𝑝 is irrational, meaning it cannot be expressed as a fraction of two integers.