Explanation

Given log₁₂(12³) = x:

Using logarithm properties, logₐ(aⁿ) = n:

x = 3

Explanation

Given b^n = b^m, for b ≠ 0:

Step 1: Analyze the equation

If b^n = b^m, then n = m, or b = 1 or b = -1 with specific conditions on n and m.

Step 2: Consider the case when b ≠ 1 and b ≠ -1

For b ≠ 0, 1, or -1, the equation holds true when n = m.

Step 3: Look for an option that satisfies n = m

Among the given options, one possible solution where n = m is when both n and m are 0, since any non-zero number to the power of 0 is 1.

The answer is: m = 0 and n = 0.

Explanation

Given sin θ > 0 and tan θ > 0:

Step 1: Determine the quadrants for sin θ > 0

sin θ > 0 in the 1st and 2nd quadrants.

Step 2: Determine the quadrants for tan θ > 0

tan θ > 0 in the 1st and 3rd quadrants.

Step 3: Find the common quadrant

The common quadrant where both conditions are met is the 1st quadrant.

Explanation

Given the roots are 1 + √5 and 1 - √5, we can find the quadratic equation using the sum and product of roots.

Step 1: Calculate the sum of roots

Sum of roots = (1 + √5) + (1 - √5) = 2

Step 2: Calculate the product of roots

Product of roots = (1 + √5)(1 - √5) = 1 - 5 = -4

Step 3: Form the quadratic equation

x² - (sum of roots)x + (product of roots) = 0

x² - 2x - 4 = 0

Explanation

Let's find the common difference (d) of the arithmetic sequence.

Step 1: Write the formula for the nth term

an = a + (n - 1)d

Step 2: Plug in the values

a = 2, n = 6, an = 27

27 = 2 + (6 - 1)d

27 = 2 + 5d

Step 3: Solve for d

25 = 5d

d = 25/5

d = 5

Explanation

A probability of 1 means that the event is certain to occur.

Probability ranges from 0 (impossible) to 1 (certain).

Explanation

Given the direct variation between x² and y, we can write:

x² = ky

where k is the constant of variation.

Step 1: Find the constant of variation k

When x = 7 and y = 49:

7² = k * 49

49 = k * 49

k = 1

Step 2: Write the equation of variation

x² = y

Step 3: Find x when y = 100

x² = 100

x = ±√100

x = ±10

The answer is: 10, -10.

Explanation

2⅔ can be written as 2^(2/3), which equals ∛(2²) = ∛4 = ³√4.

Explanation

The expression is 2x² + 8x + 8. First, factor out 2:

→ 2(x² + 4x + 4)

Now factor the quadratic:

→ x² + 4x + 4 = (x + 2)(x + 2)

So the full factorization is 2(x + 2)(x + 2).

Explanation

Let's solve the equation x/2 - x/3 = 8.

To do this, find a common denominator for the fractions, which is 6.

This gives us (3x - 2x)/6 = 8, simplifying to x/6 = 8.

Multiplying both sides by 6 gives x = 48.

So, the solution is x = 48.