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=8⋅8
16 equals 8 times 8

16=4⋅4
16 equals 4 times 4

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

Answer 1

To simplify the expression \(16 \frac{1}{4}\), we first convert it into an improper fraction.

1. Convert \(16 \frac{1}{4}\) to an improper fraction: \[ 16 \frac{1}{4} = 16 + \frac{1}{4} = \frac{64}{4} + \frac{1}{4} = \frac{65}{4} \]

Now, regarding the factors of the base \(16\) that would help in simplification, we observe the following:

• \(16 = 2^4\)
• \(16 = 4 \times 4\)
• \(16 = 8 \times 2\)

Among the options given, the most useful factorization for simplifications that involve \(2\) (often helpful for calculations and further simplifications) would be:

16 = 2⋅2⋅2⋅2 (16 equals 2 times 2 times 2 times 2)

This representation allows us to easily manipulate and simplify expressions involving powers of \(2\).

Answer 2

Given that the cube root of 27 is 3, what is 27 4/3
?(1 point)
Responses

12
12

81
81

108
108

4

Answer 3

To find \(27^{\frac{4}{3}}\), we can start by using the property of exponents that states:

\[ a^{\frac{m}{n}} = \left( a^{m} \right)^{\frac{1}{n}} \]

In this case, we can rewrite \(27^{\frac{4}{3}}\) as:

\[ 27^{\frac{4}{3}} = \left(27^{\frac{1}{3}}\right)^{4} \]

We know that the cube root of \(27\) is \(3\):

\[ 27^{\frac{1}{3}} = 3 \]

Now we can substitute this value back into the expression:

\[ \left(27^{\frac{1}{3}}\right)^{4} = (3)^{4} = 3 \times 3 \times 3 \times 3 = 81 \]

Thus, \(27^{\frac{4}{3}} = 81\).

So, the answer is:

81

Answer 4

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

37.5
37.5

125
125

75
75

625

Answer 5

To simplify \(25 \frac{3}{2}\), we first convert it into an improper fraction.

1. Convert \(25 \frac{3}{2}\) to an improper fraction: \[ 25 \frac{3}{2} = 25 + \frac{3}{2} \] To add these, we convert \(25\) into a fraction that has the same denominator: \[ 25 = \frac{50}{2} \quad \text{(since } 25 \times 2 = 50\text{)} \] Now add the fractions: \[ \frac{50}{2} + \frac{3}{2} = \frac{50 + 3}{2} = \frac{53}{2} \]

2. Simplifying \(25^{\frac{3}{2}}\): Alternatively, if the expression means \(25^{\frac{3}{2}}\), we likely need to calculate it that way.

   \[ 25^{\frac{3}{2}} = \left(25^{\frac{1}{2}}\right)^{3} \]

   The square root of \(25\) is \(5\): \[ 25^{\frac{1}{2}} = 5 \] Now raise \(5\) to the power of \(3\): \[ 5^{3} = 5 \times 5 \times 5 = 125 \]

Therefore, if you meant \(25^{\frac{3}{2}}\), the correct simplification is:

125.

If you were asking about \(25 \frac{3}{2}\) as an improper fraction, it results in \(\frac{53}{2}\), which is equivalent to \(26.5\) but among the provided options, \(125\) is indeed the correct answer for \(25^{\frac{3}{2}}\).

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

513
5 superscript Start Fraction 1 over 3 End Fraction baseline

53
5 cubed

51
5 superscript 1 baseline

35

Answer 7

To rewrite \(5 - \sqrt{3}\) using a rational exponent, we first need to clarify that \(\sqrt{3}\) can be expressed as a fractional exponent.

\(\sqrt{3}\) can be rewritten as \(3^{\frac{1}{2}}\). Therefore, the expression \(5 - \sqrt{3}\) can be written as:

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

However, if you're looking specifically for an expression involving a rational exponent related to \(5 - \sqrt{3}\) in a specific format (like one of the provided responses), there isn't a straightforward match among the answers given.

But if we consider rewriting just \(-\sqrt{3}\) as \(-3^{\frac{1}{2}}\), then the main focus should be on expressing it as a positive power like \(5^{1}\) minus something that can be captured in a rational exponent, it would still not align perfectly with the response options.

Given the options you provided, let’s analyze each:

1. \(5^{\frac{1}{3}}\): This does not relate to \(\sqrt{3}\).
2. \(5^3\): This is simply \(5\) cubed, again not related.
3. \(5^1\): This equals \(5\), and doesn't relate to \(\sqrt{3}\).
4. \(3^5\): This is \(3\) raised to the fifth power and doesn’t correlate either.

None of the provided responses directly express \(5 - \sqrt{3}\). However, if you need just the rational exponent form for \(\sqrt{3}\), the corresponding exponent would be \(3^{\frac{1}{2}}\).

If you meant to reflect just the square root of \(3\) with a rational exponent, your focus might be on the \(3^{\frac{1}{2}}\) part instead. But in the context of \(5 - \sqrt{3}\) as an overall expression, none of the options strictly apply.

Let me know if we can reformulate based on any further clarifications!

Answer 8

so what is the correct answer

Answer 9

The expression \(5 - \sqrt{3}\) itself does not directly translate to any of the provided response options. However, if the intention is to express the square root of \(3\) using a rational exponent, we could rewrite \(\sqrt{3}\) as \(3^{\frac{1}{2}}\).

However, since that does not seem relevant to the provided options, it looks like you may actually be looking for the rational exponent form of just \(-\sqrt{3}\) rather than the entire expression \(5 - \sqrt{3}\). Here's a breakdown of the options based on that realization:

1. \(5^{\frac{1}{3}}\): This indicates the cube root of 5, which is not related.
2. \(5^3\): This means \(5\) raised to the power of \(3\), irrelevant.
3. \(5^1\): This is simply \(5\), irrelevant to \(\sqrt{3}\).
4. \(3^5\): This refers to \(3\) raised to the power of \(5\), again irrelevant.

Given all that, none of the response options directly express \(5 - \sqrt{3}\) or \(\sqrt{3}\) as \(3^{\frac{1}{2}}\) in the context as it was posed.

If you're asking how to express \(\sqrt{3}\) as a rational exponent, it would be \(3^{\frac{1}{2}}\).

Therefore, the answer does not directly correspond to any of the response choices you've listed, but the most apt representation of the expression \(\sqrt{3}\) in rational exponent form is \(3^{\frac{1}{2}}\), if that translation is relevant.

If you'd like to specify whether you want just \(\sqrt{3}\) or the entire expression \(5 - \sqrt{3}\), that would help clarify! Otherwise, in terms of just rational exponents, that option is not listed.

Answer 10

How can you rewrite 8 1/2 using a root?(1 point)
Responses

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

8–√1/2
root index Start Fraction 1 over 2 End Fraction Start Root 8 End Root

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

8–√

Answer 11

To rewrite \(8^{\frac{1}{2}}\) using a root, you should recognize that the exponent \(\frac{1}{2}\) indicates the square root of the base.

Thus, \(8^{\frac{1}{2}}\) can be rewritten as:

\[ \sqrt{8} \]

Now, let's look at the provided response options to see which one correctly represents this:

1. \(12−−√8\): This doesn't correctly express \(8^{\frac{1}{2}}\).

2. \(\sqrt[2]{8}\): This correctly represents \(8^{\frac{1}{2}}\) as the square root of \(8\).

3. \(8−\sqrt{\frac{1}{2}}\): This incorrectly alters the original expression.

4. \(\sqrt[1]{8}\): This is just \(8\) since the first root is the number itself.

5. \(8−\sqrt{}\): This is an incomplete expression and doesn't apply.

From the options provided, the correct way to rewrite \(8^{\frac{1}{2}}\) (interpreted as a square root) is:

\(\sqrt{8}\)

or, if your intended notation implies \(\sqrt[2]{8}\), that is also valid, as both refer to the square root of \(8\).

If \(\sqrt[2]{8}\) is represented in your options as \(\sqrt[2]{8}\) (though it wasn't formatted in your query), that would also be correct.