Question
How can you rewrote 4 square root 21 using a ratiomal exponent?
Answers
GPT-4o mini
To rewrite \( 4 \sqrt{21} \) using a rational exponent, we can start by expressing the square root in exponent form. The square root of a number can be written as that number raised to the power of \( \frac{1}{2} \). Therefore, we have:
\[
\sqrt{21} = 21^{\frac{1}{2}}
\]
Now, substituting this back into the expression \( 4 \sqrt{21} \), we get:
\[
4 \sqrt{21} = 4 \cdot 21^{\frac{1}{2}}
\]
Next, we can express the number 4 as a power of 4, which is \( 4^{1} \). So we rewrite the expression as:
\[
4 \sqrt{21} = 4^{1} \cdot 21^{\frac{1}{2}}
\]
For further simplification, you can also combine it into a single term by considering \( 4 \) as \( 4^{\frac{2}{2}} \):
\[
4 \sqrt{21} = 4^{\frac{2}{2}} \cdot 21^{\frac{1}{2}} = \frac{4^2}{2} \cdot 21^{\frac{1}{2}}
\]
However, for a cleaner final form, it is typically left as:
\[
4 \cdot 21^{\frac{1}{2}}
\]
Thus, \( 4 \sqrt{21} \) can be expressed as:
\[
4^{1} \cdot 21^{\frac{1}{2}}
\]
This is the final form with rational exponents.
\[
\sqrt{21} = 21^{\frac{1}{2}}
\]
Now, substituting this back into the expression \( 4 \sqrt{21} \), we get:
\[
4 \sqrt{21} = 4 \cdot 21^{\frac{1}{2}}
\]
Next, we can express the number 4 as a power of 4, which is \( 4^{1} \). So we rewrite the expression as:
\[
4 \sqrt{21} = 4^{1} \cdot 21^{\frac{1}{2}}
\]
For further simplification, you can also combine it into a single term by considering \( 4 \) as \( 4^{\frac{2}{2}} \):
\[
4 \sqrt{21} = 4^{\frac{2}{2}} \cdot 21^{\frac{1}{2}} = \frac{4^2}{2} \cdot 21^{\frac{1}{2}}
\]
However, for a cleaner final form, it is typically left as:
\[
4 \cdot 21^{\frac{1}{2}}
\]
Thus, \( 4 \sqrt{21} \) can be expressed as:
\[
4^{1} \cdot 21^{\frac{1}{2}}
\]
This is the final form with rational exponents.