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Which factors of the base would help simplify the expression 1614 ?(1 point)

Responses

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

16=8⋅8
16 equals 8 times 8

16=4⋅2⋅2
16 equals 4 times 2 times 2

16=4⋅4

9 answers

To simplify the expression \( 1614 \), we need to consider the factors of \( 16 \) that may help in simplification. The most useful factorization in this context depends on \( 16 \)'s relationship to \( 1614 \).

The most relevant factorization from the given options is:

16 = 4 ⋅ 4

This representation is beneficial because \( 4 \) is a perfect square, which allows for further simplification or easier calculations, especially when dealing with square roots or powers.

Therefore, the choice that best simplifies the expression is:

16 = 4 ⋅ 4.

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

108
108

81
81

4
4

12
12

To solve for \( 2743 \), we can start by rewriting \( 2743 \) in terms of its components.

The expression \( 2743 \) can be broken down as follows:

\[ 27^{\frac{4}{3}} \]

This can be rewritten using the properties of exponents:

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

Now, we know that:

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

So we substitute that value into our expression:

\[ (3)^4 = 81 \]

Thus, the value of \( 2743 \) is:

81.

What is the correct simplification of 2532 ?(1 point)
Responses

625
625

75
75

125
125

37.5

To simplify \( 2532 \), we can recognize that it can be expressed in terms of its factors, specifically involving exponents.

The expression \( 2532 \) can be rewritten as:

\[ 25^{\frac{3}{2}} \]

This can also be expressed as:

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

We know that \( 25^{\frac{1}{2}} = 5 \), because the square root of 25 is 5. Now, substituting this into our expression, we have:

\[ (5)^3 = 125 \]

Therefore, the correct simplification of \( 2532 \) is:

125.

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

513
5 superscript Start Fraction 1 over 3 End Fraction baseline

53

To rewrite \( 5 - \sqrt{3} \) using a rational exponent, we can focus on the expression involving the square root.

The square root of \( 3 \) can be expressed as \( 3^{\frac{1}{2}} \). Therefore, we want to consider:

\[ -\sqrt{3} = -3^{\frac{1}{2}} \]

When we have \( 5 - \sqrt{3} \), it doesn't directly relate to an exponent form. However, if you're specifically referring to the negative of the square root expressed as a rational exponent, the appropriate choice from the options provided would be:

\( 3^{\frac{1}{2}} \).

Please refer to the context of the options to see how they need to be applied, but the concept stands that the square root can be expressed using a rational exponent.

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

8–√12
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–√
Start Root 8 End Root

12−−√8

To rewrite \( 812 \) using a root expression, we need to interpret the expression correctly.

The expression \( 812 \) can be understood as \( 8 \) raised to the power of \( 12 \). If you're looking to express this in terms of roots, one common way to write an exponent as a root involves using rational exponents. For instance:

\[ 8^{12} = (8^{\frac{1}{12}})^{12} = \sqrt[12]{8^{12}} \]

However, looking at your options, none of them directly translate to a valid interpretation of \( 8^{12} \).

If the goal was to express the entirety of \( 8^{12} \) as a root, the best fit from the options provided is:

root index Start Fraction 1 over 2 End Fraction Start Root 8 End Root, which represents a half-root, expressing a relationship between roots and exponents.

But if \( 812 \) was meant to represent \( 8 \) multiplied by \( 12 \) (which is often a point of misunderstanding), that would not simply convert to any root based on the options given.

Please clarify if there’s a specific interpretation intended for \( 812 \).