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Order the rational and irrational numbers from least to greatest.

11/3, PI, 3.245,√16, √8

Which point on the number line shows the position of √78?

A B C D
<----|----------|----------|------------|-------------|---------|---------|---->
6 7 8 9 10 11 12

Which sign makes the statement true?

1/8 ? 2/5

A. >

B. <

Simplify. Express your answer as a single term using exponents.

178^5*178^9

178^5 * 178^9 = ____?

Simplify the expression. (9x^4y^5)(-2x^2y^7)

Simplify the expression ∛64+3 (√2^2+5)

Simplify. Express your answer using positive exponents.

227^8/227^6=___?

The bohr radius of a hydrogen atom is 0.0000000000529. Express the bohr radius of a hydrogen atom in scientific notation.

The average weight of a blue whale is 4*10^5 pounds. The average weight of an elephant is 1*10^4 pounds. Approximately how many times heavier is a blue whale than an elephant in pounds.?

11 answers

Let's go through each problem step by step:

  1. 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:

    • \( \sqrt{8}, \pi, 3.245, \frac{11}{3}, \sqrt{16} \)

  2. 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.

  3. 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. <

  4. Simplify \( 178^5 \times 178^9 \) using exponents:

    • By the laws of exponents, \( a^m \times a^n = a^{m+n} \).
    • So, \( 178^5 \times 178^9 = 178^{5+9} = 178^{14} \).

  5. 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: \(-18x^6y^{12}\)

  6. 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.

  7. Simplify \( \frac{227^8}{227^6} \) using positive exponents:

    • By the laws of exponents, \( \frac{a^m}{a^n} = a^{m-n} \).
    • So, \( \frac{227^8}{227^6} = 227^{8-6} = 227^2 \).

  8. 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 \( 5.29 \times 10^{-11} \).

  9. 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:

  1. \( \sqrt{8}, \pi, 3.245, \frac{11}{3}, \sqrt{16} \)
  2. Point D
  3. B. <
  4. \( 178^{14} \)
  5. \(-18x^6y^{12}\)
  6. \( 25 \)
  7. \( 227^2 \)
  8. \( 5.29 \times 10^{-11} \)
  9. \( 40 \) times heavier
the only answer choices for number 6 is

13

17

21

31

Let's go through the expression in number 6 again to ensure we arrive at the correct answer based on the provided choices:

Given expression: \[ \sqrt[3]{64} + 3(\sqrt{2^2} + 5) \]

  1. Evaluate \( \sqrt[3]{64} \):

    • \( \sqrt[3]{64} = 4 \) (since \( 4 \times 4 \times 4 = 64 \)).

  2. Evaluate \( \sqrt{2^2} \):

    • \( \sqrt{2^2} = 2 \).

  3. Calculate \( 3(\sqrt{2^2} + 5) \):

    • Substitute what we found for \( \sqrt{2^2} \): \[ 3(2 + 5) = 3(7) = 21. \]

  4. Combine the two results:

    • Now add \( \sqrt[3]{64} \) and \( 3(\sqrt{2^2} + 5) \): \[ 4 + 21 = 25. \]

Since 25 is not one of the answer choices (13, 17, 21, 31), let’s verify step 3:

On examining again:

  • Correctly evaluating \( 3(\sqrt{2^2} + 5) \): \[ 3(2 + 5) = 3(7) \rightarrow 21 \text{ is indeed correct.} \]
  • Adding \( 4 + 21 \) gives us \( 25 \) which indeed does not match.

The prompt may require a reevaluation of the overall interpretation of the choices compared to complications over square root interpretations or parameter setup.

Verification step:

Let’s ensure there’s compatibility with expectations or setup misunderstanding as all solutions yield \( 25 \).

If you meant a different expression or if there’s an error in the relay, could you provide more context or clarity on what needs reevaluation? Otherwise, based on mathematical evaluation via the expression provided, \( \mathbf{25} \) remains consistent through logical interpretation of reared values utilizing roots and coefficients.

Let me know how you wish to proceed!

What is the sum of 7*10^-8 and 6*10^-8?

The amazon river releases 5.5*10^7 galls of water into the atlantic ocean every second. there are about 3.2 *10^7 seconds in a year. How many gallons are released into the ocean in one year? express your answer with the appropriate number of significant digits.

Calculate the value of the expression 35 divided by 5*7-2^3

Calculate the value of the expression:

∛110-137 +4(-5)^2

A square tile in a kitchen has an area of 121 square inches.

What is the length of one side of the square tile in inches?

Is this tile smaller or larger than one foot by one foot tile?

What is the length of the square tile in inches? ___ is the tile smaller or larger than a one foot by one foot tile?____

A. 12

B. 11

C. 9

D. Smaller

E. Larger

Which of these measurements has three significant digits?

A. 134 meters

B. 56.09 inches

C. 0.52 centimeters

D. 8009 miles

In the triangle below, the m∠1 = 64° and m∠2 = 33°, what is the m∠4?

2

1

3 (outside of the square: 4)

A. 83

B. 97

C. 116

D. 147

Let's work through each problem step by step.

  1. What is the sum of \(7 \times 10^{-8}\) and \(6 \times 10^{-8}\)? \[ 7 \times 10^{-8} + 6 \times 10^{-8} = (7 + 6) \times 10^{-8} = 13 \times 10^{-8} \] This can be expressed as: \[ 1.3 \times 10^{-7} \]

  2. Gallons of water released into the ocean in one year:

    • Water released per second: \(5.5 \times 10^7\) gallons
    • Seconds in a year: \(3.2 \times 10^7\)

    To find the total: \[ \text{Total gallons} = (5.5 \times 10^7) \times (3.2 \times 10^7) \]

    • Multiplying the coefficients: \(5.5 \times 3.2 = 17.6\)
    • For the powers of 10: \(10^7 \times 10^7 = 10^{14}\)
    • Thus, total: \[ 17.6 \times 10^{14} \] We can express this in scientific notation: \[ 1.76 \times 10^{15} \text{ gallons} \] Significant digits: The least number of significant digits in the values used is 2, so we round \(1.76\) to \(1.8\). Therefore, the final answer is: \[ 1.8 \times 10^{15} \text{ gallons} \]

  3. Calculate the value of the expression: \[ 35 \div (5 \times 7 - 2^3) \]

    • Calculate \(2^3 = 8\)
    • Now calculate \(5 \times 7 = 35\)
    • Substitute back: \[ 35 \div (35 - 8) = 35 \div 27 \] Calculate: \[ 35 \div 27 \approx 1.2963 \]

  4. Calculate the value of the expression: \[ \sqrt[3]{110} - 137 + 4 \times (-5)^2 \]

    • Calculate \((-5)^2 = 25\), so \(4 \times 25 = 100\)
    • Now substitute in: \[ \sqrt[3]{110} - 137 + 100 \]

    Approximately, \(\sqrt[3]{110} \approx 4.78\),

    Continuing: \[ 4.78 - 137 + 100 = 4.78 - 37 = -32.22 \approx -32.2 \]

  5. What is the length of one side of the square tile if the area is 121 square inches?

    • Area = side², so: \[ \text{side} = \sqrt{121} = 11 \text{ inches} \]

    Is this tile smaller or larger than a one foot by one foot tile?

    • A one-foot by one-foot tile has sides of 12 inches.
    • \(11\) inches is smaller than \(12\) inches.
    • Answer:
    • Length of the square tile in inches: \( \text{11 inches} \)
    • Is the tile smaller or larger: Smaller.

  6. Which of these measurements has three significant digits?

    • A. 134 meters (3 significant digits)
    • B. 56.09 inches (4 significant digits)
    • C. 0.52 centimeters (2 significant digits)
    • D. 8009 miles (4 significant digits)

    Hence, the answer is A. 134 meters.

  7. In the triangle where m∠1 = 64° and m∠2 = 33°, what is the m∠4?

    • The sum of angles in a triangle is \(180°\).
    • Find m∠3 (the third angle): \[ m∠3 = 180° - (m∠1 + m∠2) = 180° - (64° + 33°) = 180° - 97° = 83° \] Since m∠4 is equal to m∠3 in relation to the angles of the triangle, we have: \[ m∠4 = 83°. \] Therefore, the answer is:
    • A. 83.

Recap of Answers:

  1. \(1.3 \times 10^{-7}\)
  2. \(1.8 \times 10^{15}\) gallons
  3. \(1.2963\)
  4. \(-32.22\) (approximately)
  5. Length: 11 inches; Smaller
  6. A: 134 meters
  7. A. 83 degrees.
Can the side lengths of a triangle be 3, 6 and 9?

A. Yes

B. No

Find the value of x.

How many degrees are in a right angle? You can set the equation to equal 7x - 5+x+3=90

x=??

C 6ft.

8ft.

What is the length of the hypotenuse? If necessary, round to the nearest tenth.

c=__?

20 CM 12 CM

B

What is the length of the missing leg? If necessary, round to the nearest tenth.

B=____ centimeters

Let's solve each question step-by-step.

  1. Can the side lengths of a triangle be 3, 6, and 9?

    • To determine if three lengths can form a triangle, we must check the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side. Let's check:
      • \(3 + 6 > 9\) → \(9 > 9\) (not valid)
      • \(3 + 9 > 6\) → \(12 > 6\) (valid)
      • \(6 + 9 > 3\) → \(15 > 3\) (valid)

    Since \(3 + 6\) is not greater than \(9\), the side lengths cannot form a triangle.

    • Answer: B. No

  2. Find the value of \(x\).
    Given the equation for the angles: \[ 7x - 5 + x + 3 = 90 \] Rearranging it, we combine like terms: \[ 7x + x - 5 + 3 = 90 \ 8x - 2 = 90 \ 8x = 90 + 2 \ 8x = 92 \] Now, divide by 8: \[ x = \frac{92}{8} = 11.5 \]

    • \(x \approx 11.5\)

  3. What is the length of the hypotenuse \(c\) for a right triangle with legs of 20 cm and 12 cm?

    • Use the Pythagorean theorem: \(c^2 = a^2 + b^2\) \[ c^2 = 20^2 + 12^2 \ c^2 = 400 + 144 \ c^2 = 544 \] Now, take the square root to find \(c\): \[ c = \sqrt{544} \approx 23.3 \text{ cm} \text{ (rounding to the nearest tenth)} \]

    • Therefore, (c \approx 23.3 \text{ cm}.

  4. Find the length of the missing leg \(B\) if the hypotenuse \(c\) is given (but not specified in your question). Let’s assume you want to find one leg given the hypotenuse \(c = 20\) cm and one leg \(A = 12\) cm.

    • Using the Pythagorean theorem again: \[ c^2 = a^2 + b^2 \ (20)^2 = (12)^2 + B^2 \ 400 = 144 + B^2 \ B^2 = 400 - 144 \ B^2 = 256 \ B = \sqrt{256} = 16 \]

    • Therefore, the missing leg \(B\) is 16 centimeters.

Summary of Answers:

  1. B. No
  2. \(x = 11.5\)
  3. \(c \approx 23.3 , \text{cm}\)
  4. \(B = 16 , \text{cm}\)
Aaliyah correctly used the diagram shown to explain the pythagorean Theorem to a classmate. Which statement did aaliyah use in her explanation?

A) 5^2*5^2=3^2

B) 3^2+4^2=5^2

C) 5^2+5^2<3^2

D. 3^2+4^2>5^2

A. Choice A

B. Choice B

C. Choice C

D. Choice D

The following side lengths, in meters, were given to a carpenter to build a front porch with a triangular design. The carpenter needs to determine which set of lengths will make a triangle to be able to use it in his design. (Hint: Use the triangle inequality theorem)

Option 1: Side lengths: 3, 3, 6

Option 2: Side lengths: 6, 8, 10

Option 3: Side lengths: 4, 6, 13

Let's address each question step-by-step.

  1. Which statement did Aaliyah use in her explanation of the Pythagorean Theorem?

    • The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (\(c\)) is equal to the sum of the squares of the lengths of the other two sides (\(a\) and \(b\)). Mathematically, this is written as: \[ a^2 + b^2 = c^2 \] Hence, we look for the statement that correctly reflects this concept:
    • A) \(5^2 * 5^2 = 3^2\) (This is incorrect; it uses multiplication, which is not part of the theorem)
    • B) \(3^2 + 4^2 = 5^2\) (This is correct; \(3\), \(4\), and \(5\) can be the sides of a right triangle)
    • C) \(5^2 + 5^2 < 3^2\) (This is incorrect; it incorrectly applies comparison)
    • D) \(3^2 + 4^2 > 5^2\) (This is incorrect for the context of a right triangle; it suggests the triangle could be obtuse, but with these values, it is exactly a right triangle)

    Therefore, the correct answer is:

    • B. Choice B

  2. Determine which set of lengths can form a triangle using the triangle inequality theorem.

    • The triangle inequality theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following must hold: \[ a + b > c \] \[ a + c > b \] \[ b + c > a \]

    Let's evaluate each option:

    • Option 1: Side lengths: 3, 3, 6

      • \(3 + 3 > 6\) → \(6 \not> 6\) (not satisfied)
      • \(3 + 6 > 3\) → \(9 > 3\) (satisfied)
      • \(3 + 6 > 3\) → \(9 > 3\) (satisfied)
      • Since one of the inequalities is not satisfied, this option cannot form a triangle.

    • Option 2: Side lengths: 6, 8, 10

      • \(6 + 8 > 10\) → \(14 > 10\) (satisfied)
      • \(6 + 10 > 8\) → \(16 > 8\) (satisfied)
      • \(8 + 10 > 6\) → \(18 > 6\) (satisfied)
      • All inequalities are satisfied, so these lengths can form a triangle.

    • Option 3: Side lengths: 4, 6, 13

      • \(4 + 6 > 13\) → \(10 \not> 13\) (not satisfied)
      • \(4 + 13 > 6\) → \(17 > 6\) (satisfied)
      • \(6 + 13 > 4\) → \(19 > 4\) (satisfied)
      • Since one of the inequalities is not satisfied, this option cannot form a triangle.

Summary of Findings:

  1. The correct choice for Aaliyah's explanation is: B. Choice B
  2. The lengths that can form a triangle are from Option 2: Side lengths: 6, 8, 10.
Determine if each of the following statements is always true, sometimes true or never true.

1. The sum of the measures of two supplementary angles is 180degrees. Always true? Sometimes true? Never true?

2. The sum of complimentary angles is 180degrees. Always true? Sometimes true? Never true?

3. Two adjacent angles are complementary. Always true? Sometimes true? Never true?

the answer choices are listed at the end of the 3 options: Always true? Sometimes true? Never true?

Let's evaluate each statement one by one regarding their validity.

  1. The sum of the measures of two supplementary angles is 180 degrees.

    • Correctness: By definition, supplementary angles are two angles whose measures add up to \(180\) degrees.
    • Conclusion: This statement is Always true.

  2. The sum of complementary angles is 180 degrees.

    • Correctness: By definition, complementary angles are two angles whose measures add up to \(90\) degrees, not \(180\) degrees.
    • Conclusion: This statement is Never true.

  3. Two adjacent angles are complementary.

    • Correctness: Adjacent angles share a common side and a common vertex. These angles can be complementary (add to \(90\) degrees), but they do not have to be. For example, two adjacent angles can also sum to any value or even \(180\) degrees.
    • Conclusion: This statement is Sometimes true.

Summary of Answers:

  1. Always true
  2. Never true
  3. Sometimes true