Compare (5×3)² and Reciprocal of (4×4)⁻²: Insert > < or =

To solve this problem, let's evaluate each expression separately and then compare them:

Step 1: Evaluate $(5 \times 3)^2$

• Calculate $5 \times 3$:

$5 \times 3 = 15$

• Now, square 15:

$15^2 = 225$

Step 2: Evaluate $\frac{1}{(4 \times 4)^{-2}}$

• Calculate $4 \times 4$:

$4 \times 4 = 16$

• Apply the negative exponent rule: $a^{-n} = \frac{1}{a^n}$. Thus, $16^{-2} = \frac{1}{16^2}$.

• Calculate $16^2$:

$16^2 = 256$

• Then, take the reciprocal as indicated by the negative exponent:

$\frac{1}{16^{-2}} = \frac{1}{\frac{1}{256}} = 256$

Step 3: Compare the two results

• We have $(5 \times 3)^2 = 225$ and $\frac{1}{(4 \times 4)^{-2}} = 256$.
• Compare these values: $225 \lt 256$.

The correct sign to place between the expressions is $<$. Thus, the solution to the problem is:

$(5 \times 3)^2 \lt \frac{1}{(4 \times 4)^{-2}}$

Therefore, the correct answer is $\boxed{\lt}$.