Question
Solve x1/3<4/5 .(fractions)
Answers
GPT-4o mini
To solve the inequality \( x^{1/3} < \frac{4}{5} \), follow these steps:
1. **Cube both sides**: Since we are dealing with a cube root, we can cube both sides of the inequality to eliminate the root.
\[
(x^{1/3})^3 < \left(\frac{4}{5}\right)^3
\]
This simplifies to:
\[
x < \left(\frac{4}{5}\right)^3
\]
2. **Calculate \(\left(\frac{4}{5}\right)^3\)**:
\[
\left(\frac{4}{5}\right)^3 = \frac{4^3}{5^3} = \frac{64}{125}
\]
3. **Write the final solution**:
Thus, the solution to the inequality is:
\[
x < \frac{64}{125}
\]
In interval notation, this would be represented as:
\[
(-\infty, \frac{64}{125})
\]
1. **Cube both sides**: Since we are dealing with a cube root, we can cube both sides of the inequality to eliminate the root.
\[
(x^{1/3})^3 < \left(\frac{4}{5}\right)^3
\]
This simplifies to:
\[
x < \left(\frac{4}{5}\right)^3
\]
2. **Calculate \(\left(\frac{4}{5}\right)^3\)**:
\[
\left(\frac{4}{5}\right)^3 = \frac{4^3}{5^3} = \frac{64}{125}
\]
3. **Write the final solution**:
Thus, the solution to the inequality is:
\[
x < \frac{64}{125}
\]
In interval notation, this would be represented as:
\[
(-\infty, \frac{64}{125})
\]
mixed fraction
GPT-4o mini
To express the solution \( x < \frac{64}{125} \) as a mixed fraction, we first need to check if \( \frac{64}{125} \) can be converted into a mixed number.
Since \( 64 \) is less than \( 125 \), \( \frac{64}{125} \) is already in proper fraction form (less than 1) and cannot be expressed as a mixed number (which typically involves a whole part and a fractional part).
However, if you're looking for a decimal approximation:
\[
\frac{64}{125} = 0.512
\]
So, if you want it as a mixed fraction, it remains simply:
\[
\frac{64}{125}
\]
If we're discussing the possible whole number part that could be mixed, it would start at 0. Thus, the mixed fraction is technically \( 0 \frac{64}{125} \), but for all practical purposes, the fraction itself is sufficient.
If you need it in another specific format or additional details, please let me know!
Since \( 64 \) is less than \( 125 \), \( \frac{64}{125} \) is already in proper fraction form (less than 1) and cannot be expressed as a mixed number (which typically involves a whole part and a fractional part).
However, if you're looking for a decimal approximation:
\[
\frac{64}{125} = 0.512
\]
So, if you want it as a mixed fraction, it remains simply:
\[
\frac{64}{125}
\]
If we're discussing the possible whole number part that could be mixed, it would start at 0. Thus, the mixed fraction is technically \( 0 \frac{64}{125} \), but for all practical purposes, the fraction itself is sufficient.
If you need it in another specific format or additional details, please let me know!