Let's go through each problem step by step:
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Order the rational and irrational numbers from least to greatest:
- Find approximate values:
- \( \frac{11}{3} \approx 3.6667 \)
- \( \pi \approx 3.14159 \)
- \( 3.245 \) is already known.
- \( \sqrt{16} = 4 \)
- \( \sqrt{8} \approx 2.8284 \)
Order from least to greatest:
- \( \sqrt{8} \approx 2.8284 \)
- \( \pi \approx 3.14159 \)
- \( 3.245 \)
- \( \frac{11}{3} \approx 3.6667 \)
- \( \sqrt{16} = 4 \)
So, the ordered list is:
- \( \sqrt{8}, \pi, 3.245, \frac{11}{3}, \sqrt{16} \)
- Find approximate values:
-
Which point on the number line shows the position of \( \sqrt{78} \)?
- \( \sqrt{78} \approx 8.83 \)
- Looking at the number line:
- Point A = 6
- Point B = 7
- Point C = 8
- Point D = 9
- Therefore, \( \sqrt{78} \) is closest to point D, since it is greater than 8 but less than 9.
-
Which sign makes the statement true?
- Compare \( \frac{1}{8} \) and \( \frac{2}{5} \).
- Convert both to decimals: \( \frac{1}{8} = 0.125 \) and \( \frac{2}{5} = 0.4 \).
- Thus, \( 0.125 < 0.4 \) -> \( \frac{1}{8} < \frac{2}{5} \)
- The answer is: B. <
-
Simplify \( 178^5 \times 178^9 \) using exponents:
- By the laws of exponents, \( a^m \times a^n = a^{m+n} \).
- So, \( 178^5 \times 178^9 = 178^{5+9} = 178^{14} \).
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Simplify the expression \( (9x^4y^5)(-2x^2y^7) \):
- Multiply the coefficients and add the exponents of like bases:
- Coefficients: \( 9 \times -2 = -18 \)
- For \( x \): \( x^{4+2} = x^6 \)
- For \( y \): \( y^{5+7} = y^{12} \)
- So, the answer is: \(-18x^6y^{12}\)
-
Simplify the expression \( \sqrt[3]{64} + 3(\sqrt{2^2} + 5) \):
- \( \sqrt[3]{64} = 4 \) because \( 4 \times 4 \times 4 = 64 \).
- \( \sqrt{2^2} = 2 \).
- So, \( 3(\sqrt{2^2} + 5) = 3(2 + 5) = 3(7) = 21 \).
- Therefore, \( 4 + 21 = 25 \).
- The answer is: 25.
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Simplify \( \frac{227^8}{227^6} \) using positive exponents:
- By the laws of exponents, \( \frac{a^m}{a^n} = a^{m-n} \).
- So, \( \frac{227^8}{227^6} = 227^{8-6} = 227^2 \).
-
Express the bohr radius of a hydrogen atom in scientific notation:
- The bohr radius is \( 0.0000000000529 \).
- This can be expressed in scientific notation as \( 5.29 \times 10^{-11} \).
-
Approximately how many times heavier is a blue whale than an elephant?
- Average weight of a blue whale: \( 4 \times 10^5 \) pounds
- Average weight of an elephant: \( 1 \times 10^4 \) pounds
- To find how many times heavier: \[ \frac{4 \times 10^5}{1 \times 10^4} = 4 \times 10^{5-4} = 4 \times 10^1 = 40 \]
- Thus, a blue whale is approximately 40 times heavier than an elephant.
Here is the summary of the answers:
- \( \sqrt{8}, \pi, 3.245, \frac{11}{3}, \sqrt{16} \)
- Point D
- B. <
- \( 178^{14} \)
- \(-18x^6y^{12}\)
- \( 25 \)
- \( 227^2 \)
- \( 5.29 \times 10^{-11} \)
- \( 40 \) times heavier