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which number is irrational

A. square root of 9

B. square root of 8

C. 0.3333

D. 2/3

25 answers

An irrational number is a number that cannot be expressed as a simple fraction, meaning its decimal representation is non-repeating and non-terminating. Let's analyze each option:

A. Square root of 9 = 3, which is a rational number.

B. Square root of 8 = √8 = 2√2. Since √2 is irrational, √8 is also irrational.

C. 0.3333 is a repeating decimal (often represented as \( \frac{1}{3} \)), which is a rational number.

D. 2/3 is a fraction, which is a rational number.

Therefore, the irrational number among the listed options is B. square root of 8.

Which phrase describes the decimal expansion or the following number

4 PI

A. a non-terminating non-repeating decimal

B. a terminating decimal

C. does not have a decimal expansion

D. a repeating decimal

The number \( 4\pi \) is the product of 4 and \(\pi\), where \(\pi\) (approximately 3.14159) is an irrational number.

Since \( \pi \) is irrational, any multiple of it (like \( 4\pi \)) will also be irrational. As a result, \( 4\pi \) has a decimal expansion that is non-terminating and non-repeating.

Therefore, the correct phrase to describe the decimal expansion of \( 4\pi \) is:

A. a non-terminating non-repeating decimal.

which number has the greatest value

A. 1 2/3

B. square root of 2

C. PI/2

D. 1.5

Let's evaluate each of the options:

A. \( 1 \frac{2}{3} = 1 + \frac{2}{3} = 1.6666\ldots \) (approximately 1.67)

B. \( \sqrt{2} \approx 1.4142 \)

C. \( \frac{\pi}{2} \approx \frac{3.14159}{2} \approx 1.5708 \)

D. \( 1.5 \)

Now, let's compare these approximate values:

  • \( 1 \frac{2}{3} \approx 1.67 \)
  • \( \sqrt{2} \approx 1.4142 \)
  • \( \frac{\pi}{2} \approx 1.5708 \)
  • \( 1.5 \)

From this comparison, we can see that the greatest value is:

A. \( 1 \frac{2}{3} \) (approximately 1.67, which is greater than the other numbers).

the expression square root of 93 is a number between

A. 3 and 9

B. 8 and 9

C. 9 and 10

D. 46 and 47

To determine the range in which \( \sqrt{93} \) falls, we can calculate the squares of the integers close to \( \sqrt{93} \) and see where it fits.

  1. \( 9^2 = 81 \)
  2. \( 10^2 = 100 \)

Since \( 81 < 93 < 100 \), this indicates that:

\[ 9 < \sqrt{93} < 10 \]

Therefore, \( \sqrt{93} \) is a number between \( 9 \) and \( 10 \).

The correct option is:

C. 9 and 10.

choose all of the expressions below that are equivalent to the following

10 -^3

A. -3• -3• -3• -3• -3• -3• -3• -3• -3• -3•

B. 1000

C. 1/1000

D. 1/10^3

The expression \(10^{-3}\) represents the reciprocal of \(10^3\). Thus, it can be simplified as follows:

\[ 10^{-3} = \frac{1}{10^3} = \frac{1}{1000} \]

Now, let's evaluate each of the options:

A. \(-3 \cdot -3 \cdot -3 \cdot -3 \cdot -3 \cdot -3 \cdot -3 \cdot -3 \cdot -3 \cdot -3\) – This is \( (-3)^{10} \), which is not equivalent to \( 10^{-3} \).

B. \(1000\) – This is \(10^3\), so it is not equivalent to \(10^{-3}\).

C. \( \frac{1}{1000} \) – This is equivalent to \( 10^{-3} \).

D. \( \frac{1}{10^3} \) – This is also equivalent to \( 10^{-3} \).

Thus, the equivalent expressions to \( 10^{-3} \) are:

C. \( \frac{1}{1000} \) and D. \( \frac{1}{10^3} \).

Simplify the following
12^12 • 12-^4

A. 8^12

B. 12^16

C. 12-^48

D. 12^8

To simplify the expression \( 12^{12} \cdot 12^{-4} \), we can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \).

So, we combine the exponents:

\[ 12^{12} \cdot 12^{-4} = 12^{12 + (-4)} = 12^{12 - 4} = 12^8 \]

Thus, the simplified expression is:

D. \( 12^8 \).

Simplify

(30^3 • 32^3)^0

A. 1

B. 0

C. (30^3 • 32^3)

D. (30^4 • 32^4)

Any expression raised to the power of 0 is equal to 1, provided that the base is not zero. Therefore, we can simplify the expression \((30^3 \cdot 32^3)^0\) as follows:

\[ (30^3 \cdot 32^3)^0 = 1 \]

Thus, the correct answer is:

A. 1.

Simplify the following
(6^2) -^1

A. 36

B. 1/36

C. 6

D. 1/6

To simplify the expression \((6^2)^{-1}\), we can use the property of exponents that states \(a^{-n} = \frac{1}{a^n}\).

So we can rewrite the expression as follows:

\[ (6^2)^{-1} = \frac{1}{6^2} \]

Now calculating \(6^2\):

\[ 6^2 = 36 \]

Thus, we have:

\[ (6^2)^{-1} = \frac{1}{36} \]

Therefore, the simplified expression is:

B. \( \frac{1}{36} \).

Which equation has two possible answers

A. x^2 = 4

B. x^3 = 8

To determine which equation has two possible answers, we can analyze each equation.

A. \( x^2 = 4 \)

This equation has two solutions. We can find the solutions by taking the square root of both sides:

\[ x = \pm \sqrt{4} = \pm 2 \]

So, the solutions are \( x = 2 \) and \( x = -2 \) (two possible answers).

B. \( x^3 = 8 \)

This equation has only one real solution. To find the solution, we take the cube root of both sides:

\[ x = \sqrt[3]{8} = 2 \]

So, the solution is \( x = 2 \) (one possible answer).

Thus, the equation that has two possible answers is:

A. \( x^2 = 4 \).

Kai is at the store buying fencing for his garden he forgets the measurements he took at home but remember ps that he used 121 ft^2 of soul how much fencing will be needed for all FOUR sides of his garden

Kai will need __ feet of fencing

To determine how much fencing Kai will need for his garden, we first need to find the perimeter of the garden using the area he remembers, which is \(121 , \text{ft}^2\).

Since the area is \(121 , \text{ft}^2\), we can assume the garden is a square for simplicity (though the exact shape is not specified). The area \(A\) of a square is calculated as:

\[ A = s^2 \]

Where \(s\) is the length of one side. To find the length of one side, we can take the square root of the area:

\[ s = \sqrt{121} = 11 , \text{ft} \]

The perimeter \(P\) of a square is calculated as:

\[ P = 4s \]

Now, substituting for \(s\):

\[ P = 4 \times 11 = 44 , \text{ft} \]

Therefore, Kai will need 44 feet of fencing for all four sides of his garden.

Explain why square root of -9nis not possible burn ^2 square root of -27 is possible use comple sentences

The square root of a negative number, such as \(\sqrt{-9}\), is not possible within the realm of real numbers because there is no real number that, when squared, will yield a negative result. In basic algebra, when a real number is squared, the result is always non-negative, meaning it can either be zero or positive. Therefore, \( \sqrt{-9} \) does not have a real number solution.

However, when considering \(\sqrt{-27}\), we can express this using the concept of imaginary numbers. Any square root of a negative number can be represented using the imaginary unit \(i\), where \(i = \sqrt{-1}\).

For example, we can simplify \(\sqrt{-27}\) as follows:

\[ \sqrt{-27} = \sqrt{27} \cdot \sqrt{-1} = 3\sqrt{3} \cdot i \]

Hence, \( \sqrt{-27} \) is possible in the context of complex numbers, and it equals \(3\sqrt{3}i\), which indicates that it has a meaningful representation involving imaginary numbers.

In summary, the reason \(\sqrt{-9}\) cannot be computed in real numbers is because it is inherently negative, while \(\sqrt{-27}\) can be expressed using imaginary numbers, making it possible within complex number theory.

so the answer is?

In summary:

  1. The square root of ** \(-9\) ** is not possible in the real number system because there is no real number that, when squared, results in a negative number.

  2. **The square root of ** \(-27\) ** is possible when using imaginary numbers, as it can be expressed as ** \(3\sqrt{3}i\) , with \(i\) being the imaginary unit representing \(\sqrt{-1}\).

Thus, the simplified explanation is that while \(\sqrt{-9}\) does not exist in the realm of real numbers, \(\sqrt{-27}\) can be expressed in terms of imaginary numbers, making it a valid expression in the context of complex numbers.

thank you mamas

You're welcome! If you have any more questions or need further assistance, feel free to ask. Happy studying! 😊