Order the rational and irrational numbers from least to greatest.

Question

11/3 ,pi, 3.245,sqrt of 16 and sqrt of 8

Answer 1

To order the given numbers from least to greatest, we first need to evaluate each number:

• \( \frac{11}{3} \) = 3.6667 (approximately)
• \( \pi \) ≈ 3.1416
• \( 3.245 \) is already in decimal form.
• \( \sqrt{16} = 4 \)
• \( \sqrt{8} \) = \( \sqrt{4 \times 2} = 2\sqrt{2} \) ≈ 2.8284

Now, we can list all the approximated values:

• \( \sqrt{8} \) ≈ 2.8284
• \( \pi \) ≈ 3.1416
• \( 3.245 \)
• \( \frac{11}{3} \) ≈ 3.6667
• \( \sqrt{16} = 4 \)

Now we can order these values from least to greatest:

1. \( \sqrt{8} \) ≈ 2.8284
2. \( \pi \) ≈ 3.1416
3. \( 3.245 \)
4. \( \frac{11}{3} \) ≈ 3.6667
5. \( \sqrt{16} = 4 \)

So the final order from least to greatest is: \[ \sqrt{8}, \pi, 3.245, \frac{11}{3}, \sqrt{16} \]

Answer 2

Which sign makes the statement true?


1

8

?


2

5

(1 point)
Responses

<
<

>
>

Answer 3

To determine which sign makes the statement true, we need to compare the fractions \( \frac{1}{8} \) and \( \frac{2}{5} \).

First, let's convert both fractions to decimal form:

• \( \frac{1}{8} = 0.125 \)
• \( \frac{2}{5} = 0.4 \)

Now we can compare the two values:

• \( 0.125 < 0.4 \)

Thus, the correct sign to make the statement true is:

\[ \frac{1}{8} < \frac{2}{5} \]

Therefore, the answer is:

<

Answer 4

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

(1 point)
1785 ×1789 = ?

Answer 5

To simplify the expression \( 1785 \times 1789 \), we can express these numbers in terms of a variable. Let's denote \( x = 1787 \). Then, we can rewrite the numbers as follows:

• \( 1785 = x - 2 \)
• \( 1789 = x + 2 \)

Now we can rewrite the expression:

\[ 1785 \times 1789 = (x - 2)(x + 2) \]

This is a difference of squares, which can be simplified using the formula \( (a - b)(a + b) = a^2 - b^2 \):

\[ (x - 2)(x + 2) = x^2 - 2^2 \]

Now, substituting back for \( x \):

\[ = x^2 - 4 \] \[ = 1787^2 - 4 \]

So the simplified expression in terms of \( x \) is:

\[ 1785 \times 1789 = 1787^2 - 4 \]

If the question requires a single term using exponents, we can represent \( 1787^2 \) as is. Thus, the final answer is:

\[ 1787^2 - 4 \]

However, if necessary to express the product in the form of exponents directly without subtracting, we typically say:

\[ = 1787^2 - 4 \]

If you only meant what the result of the operation is, we can compute that as:

\[ 1785 \times 1789 = 3195125 \]

Would you like the final form strictly as an exponent term without subtraction as well? The expression as \( 1787^2 - 4 \) still stands.

Answer 6

whats the answer

Answer 7

To compute the product \( 1785 \times 1789 \) directly:

\[ 1785 \times 1789 = 3,184,365 \]

Thus, the answer is:

\[ \boxed{3,184,365} \]

Answer 8

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

Answer 9

To simplify the expression \( 1785 \times 1789 \) and express it as a single term using exponents, we can again use the method of expressing the numbers closer to a common value.

As previously mentioned, we can express:

• Let \( x = 1787 \), then:
  • \( 1785 = x - 2 \)
  • \( 1789 = x + 2 \)

Using the difference of squares:

\[ 1785 \times 1789 = (x - 2)(x + 2) = x^2 - 2^2 \]

So this simplifies to:

\[ x^2 - 4 \]

By substituting back for \( x \):

\[ = 1787^2 - 4 \]

However, if you specifically want the product as a single exponent term, the product \( 1785 \times 1789 \) corresponds to a formula without a correction term.

Using the direct statement:

\[ = (1787 - 2)(1787 + 2) = 1787^2 - 4 \]

Expressed strictly as a single term, we conclude by stating that:

\[ 1785 \times 1789 = 1787^2 - 4 \]

Thus, the answer, expressed as a single term would be:

\[ \boxed{1787^2 - 4} \]

If you're looking strictly for the format to express \( 1785 \times 1789 \) as a single term with exponents only, we would say it's simplified as \( 1787^2 - 4 \) but can't reduce it purely to a single power of a number without the subtraction term.

If you need further clarification or a different format, please let me know!

Answer 10

Simplify the expression. (2 points)
Responses
(9x*4y5*)(-2x*2y*7)

Answer 11

To simplify the expression \((9x^4y^5)(-2x^2y^7)\), we will use the distributive property of multiplication and the properties of exponents.

Step 1: Multiply the coefficients

• The coefficients are \(9\) and \(-2\): \[ 9 \times -2 = -18 \]

Step 2: Multiply the \(x\) terms

• For \(x^4\) and \(x^2\): \[ x^4 \times x^2 = x^{4+2} = x^6 \]

Step 3: Multiply the \(y\) terms

• For \(y^5\) and \(y^7\): \[ y^5 \times y^7 = y^{5+7} = y^{12} \]

Step 4: Combine all the results

Putting it all together, we have:

\[ (9x^4y^5)(-2x^2y^7) = -18x^6y^{12} \]

Thus, the simplified expression is:

\[ \boxed{-18x^6y^{12}} \]