Certainly! Let's simplify the given expression step by step.
### Expression to Simplify
[tex]\[
\frac{\left(x^1 y^{2 / 7}\right)^3}{x^{2 / 5} y^{5 / 7}}
\][/tex]
### Step 1: Simplify the Numerator
First, let's focus on the numerator [tex]\(\left(x^1 y^{2 / 7}\right)^3\)[/tex]. We need to apply the power rule, which states that [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].
#### Simplify [tex]\(x\)[/tex]-term in the Numerator
[tex]\[
(x^1)^3 = x^{1 \cdot 3} = x^3
\][/tex]
#### Simplify [tex]\(y\)[/tex]-term in the Numerator
[tex]\[
\left(y^{2 / 7}\right)^3 = y^{(2/7) \cdot 3} = y^{6/7}
\][/tex]
Thus, the numerator simplifies to:
[tex]\[
x^3 y^{6 / 7}
\][/tex]
### Step 2: Combine Numerator and Denominator
Now, rewrite the entire expression using the simplified numerator:
[tex]\[
\frac{x^3 y^{6 / 7}}{x^{2 / 5} y^{5 / 7}}
\][/tex]
### Step 3: Apply Properties of Exponents
To further simplify, we need to apply the properties of exponents, which state that [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex].
#### Simplify [tex]\(x\)[/tex]-terms
[tex]\[
\frac{x^3}{x^{2 / 5}} = x^{3 - 2/5}
\][/tex]
Calculate the exponent for [tex]\(x\)[/tex]:
[tex]\[
3 - \frac{2}{5} = 3 - 0.4 = 2.6
\][/tex]
#### Simplify [tex]\(y\)[/tex]-terms
[tex]\[
\frac{y^{6 / 7}}{y^{5 / 7}} = y^{(6 / 7) - (5 / 7)}
\][/tex]
Calculate the exponent for [tex]\(y\)[/tex]:
[tex]\[
\frac{6}{7} - \frac{5}{7} = \frac{1}{7} \approx 0.1428571428571428
\][/tex]
### Final Simplified Expression
Combining these results, the simplified form of the given expression is:
[tex]\[
x^{2.6} y^{0.1428571428571428}
\][/tex]
Thus, the fully simplified expression is:
[tex]\[
x^{2.6} y^{0.1428571428571428}
\][/tex]
