7 answers
To simplify \( 27^{1/3} \), we need to find the cube root of 27.
Since \( 27 = 3^3 \), we can rewrite the expression:
\[ 27^{1/3} = (3^3)^{1/3} \]
Using the property of exponents that states \((a^m)^n = a^{m \cdot n}\), we can simplify this further:
\[ (3^3)^{1/3} = 3^{3 \cdot (1/3)} = 3^1 = 3 \]
Thus, the simplified form of \( 27^{1/3} \) is \( \boxed{3} \).
To rewrite \( 9^{1/3} \) using a root, we recognize that \( 9^{1/3} \) represents the cube root of 9. Therefore, we can express it as follows:
\[ 9^{1/3} = \sqrt[3]{9} \]
So, \( 9^{1/3} \) rewritten using a root is \( \sqrt[3]{9} \).
To rewrite \( \sqrt{23}^4 \) using a rational number as an exponent, we first need to express the square root in exponent form. The square root of a number can be expressed as that number raised to the power of \( \frac{1}{2} \). Thus:
\[ \sqrt{23} = 23^{1/2} \]
Now, we can rewrite \( \sqrt{23}^4 \) (which is \((\sqrt{23})^4\)) as:
\[ (\sqrt{23})^4 = (23^{1/2})^4 \]
Using the property of exponents that states \((a^m)^n = a^{m \cdot n}\):
\[ (23^{1/2})^4 = 23^{(1/2) \cdot 4} = 23^{2} \]
Thus, we can express \( \sqrt{23}^4 \) as \( 23^{2} \), and the rational number used as the exponent is \( 2 \). The final answer is:
\[ 23^{2} \]
So, the rational exponent is \( 2 \).
I apologize for any confusion in my previous responses. Let's clarify the expression you provided: \( \sqrt{23}^4 \).
We can express \( \sqrt{23} \) as \( 23^{1/2} \). Thus, when we have \( \sqrt{23}^4 \), it can be rewritten using a rational exponent as follows:
\[ \sqrt{23}^4 = (23^{1/2})^4 \]
Now, applying the rule of exponents \((a^m)^n = a^{m \cdot n}\):
\[ (23^{1/2})^4 = 23^{(1/2) \cdot 4} = 23^{2} \]
However, you asked for a rational number that could be used as an exponent to rewrite \( \sqrt{23}^4 \). From the steps above, we used the exponent:
\[ \frac{4}{2} = 2 \]
Thus, to rewrite \( \sqrt{23}^4 \), the rational exponent used is \( 2 \) when simplified.
If you meant something different by "23−−√4," please let me know, and I can adjust my response accordingly. Thank you for your patience!
