Which factors of the base would help simplify the expression 16 1/4 ?(1 point)

Question

Responses

16=4⋅2⋅2
16 equals 4 times 2 times 2

16=2⋅2⋅2⋅2
16 equals 2 times 2 times 2 times 2

16=8⋅8
16 equals 8 times 8

16=4⋅4
16 equals 4 times 4

Answer 1

To simplify the expression \( 16 \frac{1}{4} \) (or \( 16.25 \)), we should look for factors of 16 that can help in calculations. The available factors of 16 from the responses are:

1. \( 16 = 4 \times 2 \times 2 \)
2. \( 16 = 2 \times 2 \times 2 \times 2 \)
3. \( 16 = 8 \times 8 \)
4. \( 16 = 4 \times 4 \)

The most straightforward way to express 16 in terms of its prime factors is:

• \( 16 = 2 \times 2 \times 2 \times 2 \)

This choice emphasizes the base of 2, which is helpful if you're also dealing with fractions or exponents, as it clearly shows that 16 is \( 2^4 \).

Thus, if we need to simplify \( 16 \frac{1}{4} \), recognizing that \( 16 = 2^4 \) is beneficial.

Therefore, the best response is:

16 = 2⋅2⋅2⋅2.

Answer 2

Given that the cube root of 27 is 3, what is 2743
?(1 point)
Responses

108
108

4
4

12
12

81

Answer 3

To calculate \( 27^{4/3} \), we can break it down using the properties of exponents:

1. First, we recognize that \( 27 = 3^3 \). Therefore, we can express \( 27^{4/3} \) as: \[ (3^3)^{4/3} \]

2. Using the property of exponents \( (a^m)^n = a^{mn} \), we get: \[ (3^3)^{4/3} = 3^{3 \cdot \frac{4}{3}} = 3^4 \]

3. Now we can calculate \( 3^4 \): \[ 3^4 = 3 \times 3 \times 3 \times 3 = 81 \]

So, the value of \( 27^{4/3} \) is:

81.

Answer 4

What is the correct simplification of 25 3/2 ?(1 point)
Responses

125
125

37.5
37.5

75
75

625

Answer 5

To simplify \( 25^{3/2} \), we can break it down as follows:

1. Recognize that \( 25 = 5^2 \). Therefore, we can express \( 25^{3/2} \) as: \[ (5^2)^{3/2} \]

2. Applying the exponent rule \((a^m)^n = a^{m \cdot n}\), we get: \[ (5^2)^{3/2} = 5^{2 \cdot \frac{3}{2}} = 5^3 \]

3. Now we can calculate \( 5^3 \): \[ 5^3 = 5 \times 5 \times 5 = 125 \]

Thus, the correct simplification of \( 25^{3/2} \) is:

125.

Answer 6

Given that the meaning of a rational exponent can be connected to the meaning of a root, how can you rewrite 5–√3 using a rational exponent?(1 point)
Responses

35
3 superscript 5 baseline

51
5 superscript 1 baseline

53
5 cubed

513

Answer 7

To rewrite \( 5 - \sqrt{3} \) using a rational exponent, we first need to understand that the square root can be expressed as a fractional exponent. Specifically, \( \sqrt{3} \) can be rewritten as \( 3^{1/2} \).

Here’s how we can express \( 5 - \sqrt{3} \) in terms of rational exponents:

\[ \sqrt{3} = 3^{1/2} \]

So, the expression \( 5 - \sqrt{3} \) can be rephrased as:

\[ 5 - 3^{1/2} \]

However, the original question might be looking for how to express \( \sqrt{3} \) specifically or how to represent the entire expression using rational exponents. The options provided don't give the complete expression, but the best rational exponent representation based on the typical conventions would be:

If the question indeed focuses only on \( \sqrt{3} \), then the answer in rational exponent form would simply be:

3^{1/2} (though this option is not listed).

If you are looking for the number assigned to the expression \( 5 - \sqrt{3} \), that isn't directly expressible in the options you provided.

If you have further context or need clarification on something specific, please let me know!

Answer 8

what option should i choose?

Answer 9

Given the options you've provided and the context of rewriting \( \sqrt{3} \) using rational exponents, it seems that the options listed are referring to different expressions that do not directly correspond to \( 5 - \sqrt{3} \).

Here’s a breakdown of the options you provided:

1. \( 3^5 \) - This is not related to \( \sqrt{3} \).
2. \( 5^1 \) - This just represents the number 5.
3. \( 5^3 \) - This represents \( 125 \) and is not relevant to \( \sqrt{3} \).
4. \( 5^{13} \) - This does not relate to \( \sqrt{3} \).

Unfortunately, none of the responses correctly correspond to expressing \( 5 - \sqrt{3} \) or \( \sqrt{3} \) in terms of rational exponents.

Since your question asks how to rewrite \( 5 - \sqrt{3} \) using a rational exponent, and if we are focusing solely on the expression for \( \sqrt{3} \), the most relevant (though still not now directly listed) would be \( 3^{1/2} \).

However, if you must choose from the given options, you might choose the option that simply represents the number \( 5 \) (which is \( 5^1 \)) because it's the only one that relates to the constant part of the expression. But be aware that it doesn't capture the \( \sqrt{3} \) part.

In summary, based on only the options provided, you should choose:

\( 5^1 \).

If you have other context or options related to \( \sqrt{3} \), please share, and I can assist further!

Answer 10

How can you rewrite 812 using a root?(1 point)
Responses

8–√
Start Root 8 End Root

8–√1
root index 1 Start Root 8 End Root

12−−√8
root index 8 Start Root Start Fraction 1 over 2 End Fraction End Root

8–√12

Answer 11

To rewrite \( 8^{12} \) using a root, you can express the exponent \( 12 \) as a fraction to represent taking a root.

The expression \( 8^{12} \) can be rewritten as:

\[ 8^{12} = (8^{1/2})^{24} \quad \text{(using the root and expressing the exponent as a fraction)} \]

However, to directly choose from your provided options that incorporate roots, we specifically represent \( 8^{12} \) in terms of a root.

From the options given:

1. \( \sqrt{8} \) — This is simply the square root of 8, which is not a correct representation of \( 8^{12} \).
2. \( \sqrt[1]{8} \) — This is the first root of 8, which is just 8; not a correct representation of \( 8^{12} \).
3. \( \sqrt[8]{\frac{1}{2}} \) — This indicates a complex root and not simply related to \( 8^{12} \).
4. \( \sqrt{12} \) — This is the square root of 12; not a correct representation of \( 8^{12} \).

None of the options appears to directly represent \( 8^{12} \) correctly. However, if you are looking to express in terms of roots more generally, the expression \( 8^{12} \) will be best understood as:

\[ 8^{12} = (8^{12/2})^2 = (8^6)^2 = \sqrt{8^{12}} \]

If you must select based from the options that relate to the general idea of roots, the closest choice could perhaps be interpreted as needing to rewrite under general relationships, but there is no correct representation of \( 8^{12} \) in those options.

If this is about simplification or basic options for using roots, I would suggest clarifying the context; otherwise, none of the options accurately represent \( 8^{12} \).