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Answer :
- Apply the quotient rule for exponents: $\frac{a^m}{a^n} = a^{m-n}$.
- Subtract the exponents: $\frac{17}{15} - \frac{14}{15} = \frac{3}{15}$.
- Simplify the fraction: $\frac{3}{15} = \frac{1}{5}$.
- The simplified expression is: $\boxed{y^{\frac{1}{5}}}$.
### Explanation
1. Understanding the problem and the approach
We are given the expression $\frac{y^{\frac{17}{15}}}{y^{\frac{14}{15}}}$ and asked to simplify it. To do this, we will use the quotient rule for exponents, which states that when dividing terms with the same base, we subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$.
2. Applying the quotient rule
Applying the quotient rule to our expression, we get:
$$\frac{y^{\frac{17}{15}}}{y^{\frac{14}{15}}} = y^{\frac{17}{15} - \frac{14}{15}}$$
3. Simplifying the exponent
Now, we need to subtract the exponents:
$$\frac{17}{15} - \frac{14}{15} = \frac{17 - 14}{15} = \frac{3}{15}$$
We can simplify the fraction $\frac{3}{15}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 3:
$$\frac{3}{15} = \frac{3 \div 3}{15 \div 3} = \frac{1}{5}$$
So, the exponent simplifies to $\frac{1}{5}$.
4. Final result
Therefore, the simplified expression is:
$$y^{\frac{1}{5}}$$
This can also be written as the fifth root of y:
$$\sqrt[5]{y}$$
5. Conclusion
The simplified form of the given expression is $y^{\frac{1}{5}}$.
### Examples
Imagine you are calculating the growth of a plant where the growth rate is expressed as a fractional exponent. Simplifying expressions with fractional exponents, like the one we just did, allows you to easily understand and compare different growth rates. For example, if you have two plants with growth rates of $y^{\frac{17}{15}}$ and $y^{\frac{14}{15}}$, simplifying the expression helps you determine the relative difference in their growth.
- Subtract the exponents: $\frac{17}{15} - \frac{14}{15} = \frac{3}{15}$.
- Simplify the fraction: $\frac{3}{15} = \frac{1}{5}$.
- The simplified expression is: $\boxed{y^{\frac{1}{5}}}$.
### Explanation
1. Understanding the problem and the approach
We are given the expression $\frac{y^{\frac{17}{15}}}{y^{\frac{14}{15}}}$ and asked to simplify it. To do this, we will use the quotient rule for exponents, which states that when dividing terms with the same base, we subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$.
2. Applying the quotient rule
Applying the quotient rule to our expression, we get:
$$\frac{y^{\frac{17}{15}}}{y^{\frac{14}{15}}} = y^{\frac{17}{15} - \frac{14}{15}}$$
3. Simplifying the exponent
Now, we need to subtract the exponents:
$$\frac{17}{15} - \frac{14}{15} = \frac{17 - 14}{15} = \frac{3}{15}$$
We can simplify the fraction $\frac{3}{15}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 3:
$$\frac{3}{15} = \frac{3 \div 3}{15 \div 3} = \frac{1}{5}$$
So, the exponent simplifies to $\frac{1}{5}$.
4. Final result
Therefore, the simplified expression is:
$$y^{\frac{1}{5}}$$
This can also be written as the fifth root of y:
$$\sqrt[5]{y}$$
5. Conclusion
The simplified form of the given expression is $y^{\frac{1}{5}}$.
### Examples
Imagine you are calculating the growth of a plant where the growth rate is expressed as a fractional exponent. Simplifying expressions with fractional exponents, like the one we just did, allows you to easily understand and compare different growth rates. For example, if you have two plants with growth rates of $y^{\frac{17}{15}}$ and $y^{\frac{14}{15}}$, simplifying the expression helps you determine the relative difference in their growth.
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