Explanation

Given the pair of equations:

1. x²y - 2y = 42

2. x + 2 = 5

From equation (2), we get:

x = 5 - 2

x = 3

Now, substitute x = 3 into equation (1):

3²y - 2y = 42

9y - 2y = 42

7y = 42

y = 6

Explanation

Let's calculate the value of the function:

f(x) = x³ + 3x + 9

f(-1) = (-1)³ + 3(-1) + 9

= -1 - 3 + 9

= 5

Explanation

Let's solve for x:

4x - 15 = 1

Add 15 to both sides:

4x = 1 + 15

4x = 16

Divide both sides by 4:

x = 16/4

x = 4

Explanation

The correct logarithmic form of x = a^y is:

logₐx = y

This is because logarithms and exponentials are inverse operations.

Explanation

Let's simplify the expression:

∜256a⁴b⁴ = ∜(4⁴ × a⁴ × b⁴)

= ∜(4a × 4a × 4a × 4a × b × b × b × b)

= 4ab

Explanation

Let's simplify the expression:

(((3)²)²)² = (3²)⁴ = 3⁸

Now, divide by 9 (which is 3²):

3⁸ / 3² = 3⁶

Explanation

Let's evaluate each option:

1. sin² 60⁰ + cos² 60⁰ = 1

sin 60⁰ = √3/2, cos 60⁰ = 1/2

(√3/2)² + (1/2)² = 3/4 + 1/4 = 1

2. sin² 45⁰ + cos² 45⁰ = 1

sin 45⁰ = cos 45⁰ = 1/√2

(1/√2)² + (1/√2)² = 1/2 + 1/2 = 1

3. sin² 45⁰ − cos² 45⁰ = 0

(1/√2)² - (1/√2)² = 1/2 - 1/2 = 0

Explanation

The expression x³ + 27 can be factored using the sum of cubes formula:

x³ + 27 = x³ + 3³

= (x + 3)(x² - 3x + 9)

Explanation

Given:

x / 4 = y / 2

Multiply both sides by 4:

x = 2y

Explanation

Given:

(3z + 4z + 5z) / 2 = 36

Combine like terms:

12z / 2 = 36

6z = 36

z = 36 / 6

z = 6