The HCF of 48, 60, and 72 is 12, as it is the largest number that divides all three.

The HCF of 48, 64, and 80 is 16, as it is the largest number that divides all three.

The LCM of 6 and 8 is 24, as it is the smallest number that is a multiple of both.

The LCM of 9 and 15 is 45, as it is the smallest number that is a multiple of both.

Add 12 to both sides: 3x^2 = 12. Divide by 3: x^2 = 4. Thus, x = ±2.

Using the quadratic formula x = [-b ± √(b² - 4ac)] / 2a gives x = [-8 ± √(64 - 48)] / 8 = [-8 ± 4] / 8.

Factoring gives (4x - 3)(x - 5) = 0, so x = 3 or x = 5.

Subtract 3x from both sides: 2x - 2 = 6. Add 2: 2x = 8. Divide by 2: x = 4.

Distributing gives 5x - 6 + 2x = 4. Combining like terms: 7x - 6 = 4. Thus, 7x = 10, x = 10/7.

This is a perfect square: (x + 3)^2 = 0, so x = -3.

Using the quadratic formula x = [-b ± √(b² - 4ac)] / 2a, we find x = [-3 ± √(9 + 40)] / 4.

Dividing the equation by 2 gives x^2 + 4x + 3 = 0, which factors to (x + 1)(x + 3) = 0.

16 = 2^(4), so x = 4.

32 = 2^(5), so x = 5.

81 = 3^(4), so x = 4.

Subtract 2 from both sides: 4x = 16. Divide by 4: x = 4.

Add 16 to both sides: 4x^2 = 16. Divide by 4: x^2 = 4. Thus, x = ±2.

Subtract 3x from both sides: 2x - 2 = 6. Add 2: 2x = 8. Divide by 2: x = 4.

Subtract 3x from both sides: 2x - 7 = 5. Add 7 to both sides: 2x = 12. Divide by 2: x = 6.

Subtract 3x from both sides: 2x - 7 = 9. Add 7: 2x = 16. Divide by 2: x = 8.

Add 20 to both sides: 5x^2 = 20. Divide by 5: x^2 = 4. Thus, x = ±2.

a/b = 2^(3)/2^(2) = 2^(3-2) = 2^(1).

a^(b+1) = 3^(2+1) = 3^3 = 27.

a^4 = (3^(1/2))^4 = 3^2 = 9.

b^6 = (2^(1/3))^6 = 2^2 = 4.

log2(8) = log2(2^3) = 3.

27 = 3^3, so x = 3.

log3(9) = 2 and log3(3) = 1, so x = 2 + 1 = 3.

x = 16^(1/4) = 2.

If the HCF is 1, the numbers are coprime, meaning they have no common factors other than 1.