answer:To find the numbers of the form ( 36^k - 5^l ) that have the smallest absolute value, we need to experiment with various values of ( k ) and ( l ). We should start with small values for ( k ) and ( l ) and iterate to see which combination gives the smallest absolute value.Let's begin by setting a reasonable range for ( k ) and ( l ) (e.g., from 1 to 10) and calculate ( 36^k - 5^l ). Then, we will determine the smallest absolute value from these calculations.Here's the approach in Python code:1. Set a range for ( k ) and ( l ).2. Calculate ( 36^k - 5^l ) for each combination of ( k ) and ( l ).3. Keep track of the smallest absolute value and corresponding ( k ) and ( l ).Let's implement this and find the smallest value.```python# Define the range for k and l k_range = range(1, 11) l_range = range(1, 11) # Initialize variable to keep track of the smallest absolute value and the corresponding k and l smallest_abs_value = float('inf') best_k = None best_l = None # Iterate over the range of k and l for k in k_range: for l in l_range: # Calculate the value of 36^k - 5^l value = 36k - 5l # Check the absolute value abs_value = abs(value) # Update the smallest absolute value and corresponding k and l if needed if abs_value < smallest_abs_value: smallest_abs_value = abs_value best_k = k best_l = l # Print the result print((smallest_abs_value, best_k, best_l))``````output(11, 1, 2)```The smallest absolute value of the form ( 36^k - 5^l ) is 11, which occurs when ( k = 1 ) and ( l = 2 ).Thus, the smallest absolute value is:[boxed{11}]

answer:To solve this problem, we need to follow a few logical steps. Let's break down the problem and solve it step-by-step.1. Define the original area of the rectangle: Let's denote the original length by ( L ) and the original width by ( W ). The original area ( A_{text{original}} ) is given by: [ A_{text{original}} = L times W ]2. Compute the new dimensions of the rectangle: - The length is increased by 20%, so the new length ( L_{text{new}} ) is: [ L_{text{new}} = L times 1.20 ] - The width is increased by 10%, so the new width ( W_{text{new}} ) is: [ W_{text{new}} = W times 1.10 ]3. Calculate the new area of the rectangle: The new area ( A_{text{new}} ) is: [ A_{text{new}} = L_{text{new}} times W_{text{new}} = (L times 1.20) times (W times 1.10) ]4. Determine the percentage increase in area: The percentage increase in area is given by: [ text{Percentage Increase} = left( frac{A_{text{new}} - A_{text{original}}}{A_{text{original}}} right) times 100 ]Let's implement this in Python using SymPy to ensure accuracy.```pythonimport sympy as sp# Define the original length and width as symbolsL, W = sp.symbols('L W')# Define the original areaA_original = L * W# Calculate the new length and widthL_new = L * 1.20W_new = W * 1.10# Calculate the new areaA_new = L_new * W_new# Calculate the percentage increase in the areapercentage_increase = ((A_new - A_original) / A_original) * 100# Simplify the percentage increase expressionpercentage_increase_simplified = sp.simplify(percentage_increase)# Output the resultprint(percentage_increase_simplified)``````output32.0000000000000```The percentage increase in the area of the rectangle when its length is increased by 20% and its width is increased by 10% is (boxed{32%}).This result means that the area of the rectangle will increase by 32% under the given conditions.

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