Question

Find the least common denominator. $$\frac{3}{4 x}, \frac{1}{6 x^{2}}, \frac{1}{8 x^{2}}$$

Short answer

The least common denominator for \(\frac{3}{4 x}, \frac{1}{6 x^{2}}, \frac{1}{8 x^{2}}\) is \(24x^2\).

Step-by-step solution

List Down the Factors

Start by listing down the prime factors of each denominator. \[4x = 2^2 \cdot x\], \[6x^2 = 2 \cdot 3 \cdot x^2\], \[8x^2 = 2^3\cdot x^2\].

Identify Highest Powers

The next step is to identify the highest power of each prime number across the denominators. This is given as \[2^3, 3^1, x^2\]

Calculate the Least Common Denominator

Finally, multiply these highest powers together to obtain the least common denominator: \[2^3 \cdot 3^1 \cdot x^2 = 24x^2\]