What is the largest number among the following?

Correct Option (c)

To determine the largest number, each expression must be evaluated. For option (c), the expression is ${\left(\frac{1}{3}\right)}^{-4}$. Utilizing the property $a^{-n}=\frac{1}{a^n}$ or ${\left(\frac{1}{x}\right)}^{-n}=x^n$, this simplifies to $3^4$. Calculating $3^4$ yields \(3 \times 3 \times 3 \times 3 = 81\). This value represents the maximum among the given options.

Incorrect Options:

The other options evaluate as follows:

• For option (a), ${\left(\frac{1}{2}\right)}^{-6}$ simplifies to $2^6$. Calculating $2^6$ gives \(2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\).
• For option (b), ${\left(\frac{1}{4}\right)}^{-3}$ simplifies to $4^3$. Calculating $4^3$ gives \(4 \times 4 \times 4 = 64\).
• For option (d), ${\left(\frac{1}{6}\right)}^{-2}$ simplifies to $6^2$. Calculating $6^2$ gives \(6 \times 6 = 36\).

Comparing the evaluated values (64, 64, 81, 36), it is evident that 81 is the largest number, making options (a), (b), and (d) incorrect because their values are smaller than 81.