Question
which factors of the base would help simplify the expression 16 1/4
16= 4x4
16=8x8
16=4x2x2
16=2x2x2x2
16= 4x4
16=8x8
16=4x2x2
16=2x2x2x2
Answers
GPT-4o mini
To simplify the expression \( 16 \frac{1}{4} \), it can be helpful to first convert the mixed number into an improper fraction:
\[
16 \frac{1}{4} = 16 + \frac{1}{4} = \frac{16 \times 4 + 1}{4} = \frac{64 + 1}{4} = \frac{65}{4}
\]
Now let's use the factors of the base (16) for simplification purposes. The factors you provided can be listed as:
1. \( 16 = 4 \times 4 \)
2. \( 16 = 8 \times 8 \)
3. \( 16 = 4 \times 2 \times 2 \)
4. \( 16 = 2 \times 2 \times 2 \times 2 \)
Since we're considering \( 16 \frac{1}{4} \), let's rewrite \( 16 \) as \( 2^4 \):
\[
\frac{65}{4} = \frac{65}{2^2}
\]
From this point, you can use the Prime Factorization method or express \( 16 \) in its factor forms but ultimately, given that \( 16 \) in the given expressions are just different factorizations of \( 2^4 \), the use of these doesn't simplify the fraction further. Thus, we will just express \( 16 \) generally.
That said, the expression remains simplified as:
\[
16 \frac{1}{4} = \frac{65}{4}
\]
In terms of factors of 16 simplifying the expression, none of them directly contribute to further simplification of the fraction. The important base simplification just shows the relationship that \( 16 = 2^4 \) can be useful if you're working on problems requiring prime factorization, but for standalone simplification of \( 16 \frac{1}{4} \), it concludes as \( \frac{65}{4} \).
\[
16 \frac{1}{4} = 16 + \frac{1}{4} = \frac{16 \times 4 + 1}{4} = \frac{64 + 1}{4} = \frac{65}{4}
\]
Now let's use the factors of the base (16) for simplification purposes. The factors you provided can be listed as:
1. \( 16 = 4 \times 4 \)
2. \( 16 = 8 \times 8 \)
3. \( 16 = 4 \times 2 \times 2 \)
4. \( 16 = 2 \times 2 \times 2 \times 2 \)
Since we're considering \( 16 \frac{1}{4} \), let's rewrite \( 16 \) as \( 2^4 \):
\[
\frac{65}{4} = \frac{65}{2^2}
\]
From this point, you can use the Prime Factorization method or express \( 16 \) in its factor forms but ultimately, given that \( 16 \) in the given expressions are just different factorizations of \( 2^4 \), the use of these doesn't simplify the fraction further. Thus, we will just express \( 16 \) generally.
That said, the expression remains simplified as:
\[
16 \frac{1}{4} = \frac{65}{4}
\]
In terms of factors of 16 simplifying the expression, none of them directly contribute to further simplification of the fraction. The important base simplification just shows the relationship that \( 16 = 2^4 \) can be useful if you're working on problems requiring prime factorization, but for standalone simplification of \( 16 \frac{1}{4} \), it concludes as \( \frac{65}{4} \).
just pick the correct answer
GPT-4o mini
The correct factorization of 16 that would help simplify the expression \( 16 \frac{1}{4} \) is:
**16 = 4 × 4**
**16 = 4 × 4**
given that the cube root of 27 is 3, what is 27 4/3?
GPT-4o mini
To simplify \( 27^{\frac{4}{3}} \), we can use the property of exponents, which states that:
\[
a^{m/n} = \sqrt[n]{a^m}
\]
In this case, we can break it down as follows:
\[
27^{\frac{4}{3}} = \left(27^{\frac{1}{3}}\right)^{4}
\]
Since we know that the cube root of 27 is 3:
\[
27^{\frac{1}{3}} = 3
\]
Now we can substitute this back into our expression:
\[
27^{\frac{4}{3}} = 3^{4}
\]
Calculating \( 3^4 \):
\[
3^4 = 81
\]
Therefore,
\[
27^{\frac{4}{3}} = 81
\]
\[
a^{m/n} = \sqrt[n]{a^m}
\]
In this case, we can break it down as follows:
\[
27^{\frac{4}{3}} = \left(27^{\frac{1}{3}}\right)^{4}
\]
Since we know that the cube root of 27 is 3:
\[
27^{\frac{1}{3}} = 3
\]
Now we can substitute this back into our expression:
\[
27^{\frac{4}{3}} = 3^{4}
\]
Calculating \( 3^4 \):
\[
3^4 = 81
\]
Therefore,
\[
27^{\frac{4}{3}} = 81
\]
what is the correct simplification of 25 3/2
GPT-4o mini
To simplify \( 25 \frac{3}{2} \), we first convert it into an improper fraction:
\[
25 \frac{3}{2} = 25 + \frac{3}{2} = \frac{25 \times 2}{2} + \frac{3}{2} = \frac{50 + 3}{2} = \frac{53}{2}
\]
However, if you are looking to simplify \( 25^{\frac{3}{2}} \) (not the mixed number but the exponent), we can interpret this differently.
**Calculating \( 25^{\frac{3}{2}} \)**:
1. First, recognize that \( 25 = 5^2 \).
2. Thus, we can express \( 25^{\frac{3}{2}} \) as:
\[
(5^2)^{\frac{3}{2}}
\]
3. Now, apply the exponent rule \( (a^m)^n = a^{m \cdot n} \):
\[
5^{2 \cdot \frac{3}{2}} = 5^3
\]
4. Now calculate \( 5^3 \):
\[
5^3 = 125
\]
So, in the case of simplification, we find that:
\[
25^{\frac{3}{2}} = 125
\]
If you meant something else by \( 25 \frac{3}{2} \), please clarify!
\[
25 \frac{3}{2} = 25 + \frac{3}{2} = \frac{25 \times 2}{2} + \frac{3}{2} = \frac{50 + 3}{2} = \frac{53}{2}
\]
However, if you are looking to simplify \( 25^{\frac{3}{2}} \) (not the mixed number but the exponent), we can interpret this differently.
**Calculating \( 25^{\frac{3}{2}} \)**:
1. First, recognize that \( 25 = 5^2 \).
2. Thus, we can express \( 25^{\frac{3}{2}} \) as:
\[
(5^2)^{\frac{3}{2}}
\]
3. Now, apply the exponent rule \( (a^m)^n = a^{m \cdot n} \):
\[
5^{2 \cdot \frac{3}{2}} = 5^3
\]
4. Now calculate \( 5^3 \):
\[
5^3 = 125
\]
So, in the case of simplification, we find that:
\[
25^{\frac{3}{2}} = 125
\]
If you meant something else by \( 25 \frac{3}{2} \), please clarify!