Question

1. is the sum of the areas of two smaller squares equal to the area of a large square if the side - lengths of the squares are 8 feet, 4 feet, and 3 feet? note that the area of a square is s², where s is the side - length. explain.
2. lexi bought a new car. she drove 5² miles in the first month that she owned the car and 4² miles in the second month that she owned the car. how many miles did lexi drive in all during the first two months that she owned the car?
3. communicate and justify. is 0.3⁴ equal to 0.9? explain.
4. what are two ways that you can represent 27 using 3?

Explanation:

1. For the problem about the areas of squares:

• # Explanation:
• ## Step1: Calculate the area of the first - smaller square
• The side - length of the first smaller square is \(s_1 = 8\) feet. Using the formula \(A = s^{2}\), the area \(A_1=s_1^{2}=8^{2}=64\) square feet.
• ## Step2: Calculate the area of the second - smaller square
• The side - length of the second smaller square is \(s_2 = 3\) feet. Using the formula \(A = s^{2}\), the area \(A_2=s_2^{2}=3^{2}=9\) square feet.
• ## Step3: Calculate the sum of the areas of the two smaller squares
• \(A = A_1+A_2=64 + 9=73\) square feet. Let the side - length of the large square be \(s\). Then \(s^{2}=73\), and \(s=\sqrt{73}\approx8.54\) feet.
• # Answer: The side - length of the large square is \(\sqrt{73}\approx8.54\) feet.

1. For the problem about Lexi's driving distance:

• # Explanation:
• ## Step1: Calculate the distance driven in the first month
• Lexi drove \(5^{2}\) miles in the first month. Since \(5^{2}=25\) miles.
• ## Step2: Calculate the distance driven in the second month
• Lexi drove \(4^{2}\) miles in the second month. Since \(4^{2}=16\) miles.
• ## Step3: Calculate the total distance driven in the first two months
• The total distance \(d=5^{2}+4^{2}=25 + 16=41\) miles.
• # Answer: 41 miles

1. For the problem about \(0.3^{4}\) and \(0.09^{2}\):

• # Explanation:
• ## Step1: Calculate \(0.3^{4}\)
• \(0.3^{4}=0.3\times0.3\times0.3\times0.3=(0.3\times0.3)\times(0.3\times0.3)=0.09\times0.09 = 0.0081\).
• ## Step2: Calculate \(0.09^{2}\)
• \(0.09^{2}=0.09\times0.09 = 0.0081\).
• So \(0.3^{4}=0.09^{2}\) because \(0.3^{4}=(0.3\times0.3)\times(0.3\times0.3)=0.09\times0.09 = 0.09^{2}\).
• # Answer: Yes, \(0.3^{4}\) is equal to \(0.09^{2}\) because \(0.3^{4}=(0.3\times0.3)\times(0.3\times0.3)=0.09\times0.09 = 0.09^{2}\).

1. For the problem about representing 27 using 3:

• # Explanation:
• ## Step1: First way
• \(3^{3}=3\times3\times3 = 27\).
• ## Step2: Second way
• \(3\times9=27\), and since \(9 = 3\times3\), we can write \(27=3\times(3\times3)\).
• # Answer: \(3^{3}\) and \(3\times(3\times3)\)

Answer:

1. For the problem about the areas of squares:

• # Explanation:
• ## Step1: Calculate the area of the first - smaller square
• The side - length of the first smaller square is \(s_1 = 8\) feet. Using the formula \(A = s^{2}\), the area \(A_1=s_1^{2}=8^{2}=64\) square feet.
• ## Step2: Calculate the area of the second - smaller square
• The side - length of the second smaller square is \(s_2 = 3\) feet. Using the formula \(A = s^{2}\), the area \(A_2=s_2^{2}=3^{2}=9\) square feet.
• ## Step3: Calculate the sum of the areas of the two smaller squares
• \(A = A_1+A_2=64 + 9=73\) square feet. Let the side - length of the large square be \(s\). Then \(s^{2}=73\), and \(s=\sqrt{73}\approx8.54\) feet.
• # Answer: The side - length of the large square is \(\sqrt{73}\approx8.54\) feet.

1. For the problem about Lexi's driving distance:

• # Explanation:
• ## Step1: Calculate the distance driven in the first month
• Lexi drove \(5^{2}\) miles in the first month. Since \(5^{2}=25\) miles.
• ## Step2: Calculate the distance driven in the second month
• Lexi drove \(4^{2}\) miles in the second month. Since \(4^{2}=16\) miles.
• ## Step3: Calculate the total distance driven in the first two months
• The total distance \(d=5^{2}+4^{2}=25 + 16=41\) miles.
• # Answer: 41 miles

1. For the problem about \(0.3^{4}\) and \(0.09^{2}\):

• # Explanation:
• ## Step1: Calculate \(0.3^{4}\)
• \(0.3^{4}=0.3\times0.3\times0.3\times0.3=(0.3\times0.3)\times(0.3\times0.3)=0.09\times0.09 = 0.0081\).
• ## Step2: Calculate \(0.09^{2}\)
• \(0.09^{2}=0.09\times0.09 = 0.0081\).
• So \(0.3^{4}=0.09^{2}\) because \(0.3^{4}=(0.3\times0.3)\times(0.3\times0.3)=0.09\times0.09 = 0.09^{2}\).
• # Answer: Yes, \(0.3^{4}\) is equal to \(0.09^{2}\) because \(0.3^{4}=(0.3\times0.3)\times(0.3\times0.3)=0.09\times0.09 = 0.09^{2}\).

1. For the problem about representing 27 using 3:

• # Explanation:
• ## Step1: First way
• \(3^{3}=3\times3\times3 = 27\).
• ## Step2: Second way
• \(3\times9=27\), and since \(9 = 3\times3\), we can write \(27=3\times(3\times3)\).
• # Answer: \(3^{3}\) and \(3\times(3\times3)\)