Question
According to exponent rules, what is the rational exponent in the equitation to ^3 5 enter your response in fraction form.
Answers
Answer
Use the properties of exponents to generate an equivalent expression for ^3 8 / 2
Answered by
GPT-4o mini
The expression \( 5^{\frac{1}{3}} \) indicates that the rational exponent is \( \frac{1}{3} \).
In general, the expression \( a^{\frac{m}{n}} \) represents the \( n \)-th root of \( a \) raised to the power of \( m \). In the case of \( 5^{\frac{1}{3}} \), this means the cube root of \( 5 \).
So, the rational exponent in your expression is \( \frac{1}{3} \).
In general, the expression \( a^{\frac{m}{n}} \) represents the \( n \)-th root of \( a \) raised to the power of \( m \). In the case of \( 5^{\frac{1}{3}} \), this means the cube root of \( 5 \).
So, the rational exponent in your expression is \( \frac{1}{3} \).
Answered by
GPT-4o mini
To simplify the expression \(\frac{8^3}{2}\) using the properties of exponents, we start by rewriting the expression:
1. Recognize that \(8\) can be expressed as \(2^3\):
\[
8 = 2^3
\]
2. Substitute \(8\) in the expression:
\[
\frac{(2^3)^3}{2}
\]
3. Apply the power of a power property, which states that \((a^m)^n = a^{mn}\):
\[
\frac{2^{3 \cdot 3}}{2} = \frac{2^9}{2}
\]
4. Use the quotient of powers property, which states that \(\frac{a^m}{a^n} = a^{m-n}\):
\[
2^{9-1} = 2^8
\]
Thus, an equivalent expression for \(\frac{8^3}{2}\) is \(2^8\).
1. Recognize that \(8\) can be expressed as \(2^3\):
\[
8 = 2^3
\]
2. Substitute \(8\) in the expression:
\[
\frac{(2^3)^3}{2}
\]
3. Apply the power of a power property, which states that \((a^m)^n = a^{mn}\):
\[
\frac{2^{3 \cdot 3}}{2} = \frac{2^9}{2}
\]
4. Use the quotient of powers property, which states that \(\frac{a^m}{a^n} = a^{m-n}\):
\[
2^{9-1} = 2^8
\]
Thus, an equivalent expression for \(\frac{8^3}{2}\) is \(2^8\).
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