Let's simplify each expression step-by-step and express the results with positive exponents.
### Part (i): [tex]\(\left(x^{-3}\right)^4\)[/tex]
To simplify this expression, we use the power of a power property of exponents, which states:
[tex]\[ (a^m)^n = a^{m \cdot n} \][/tex]
Here, [tex]\(a = x\)[/tex], [tex]\(m = -3\)[/tex], and [tex]\(n = 4\)[/tex]. So,
[tex]\[
\left(x^{-3}\right)^4 = x^{-3 \cdot 4} = x^{-12}
\][/tex]
To express the result with a positive exponent, we use the property of exponents that states:
[tex]\[ a^{-m} = \frac{1}{a^m} \][/tex]
Therefore,
[tex]\[
x^{-12} = \frac{1}{x^{12}}
\][/tex]
So, the simplified expression for (i) is:
[tex]\[
\left(x^{-3}\right)^4 = \frac{1}{x^{12}}
\][/tex]
### Part (ii): [tex]\(3 x^{\frac{1}{6}} \times 3 x^{-\frac{7}{6}}\)[/tex]
First, we combine the coefficients:
[tex]\[
3 \times 3 = 9
\][/tex]
Next, we combine the exponents of [tex]\(x\)[/tex] using the product of powers property, which states:
[tex]\[ a^m \times a^n = a^{m + n} \][/tex]
Here, we have [tex]\(x^{\frac{1}{6}}\)[/tex] and [tex]\(x^{-\frac{7}{6}}\)[/tex], so we add the exponents:
[tex]\[
\frac{1}{6} + \left(-\frac{7}{6}\right) = \frac{1}{6} - \frac{7}{6} = -\frac{6}{6} = -1
\][/tex]
Thus,
[tex]\[
x^{\frac{1}{6}} \times x^{-\frac{7}{6}} = x^{-1}
\][/tex]
Now we have:
[tex]\[
9 \cdot x^{-1}
\][/tex]
To express this with a positive exponent, we use the property [tex]\(a^{-m} = \frac{1}{a^m}\)[/tex]:
[tex]\[
9 \cdot x^{-1} = 9 \cdot \frac{1}{x} = \frac{9}{x}
\][/tex]
Therefore, the simplified expression for (ii) is:
[tex]\[
3 x^{\frac{1}{6}} \times 3 x^{-\frac{7}{6}} = \frac{9}{x}
\][/tex]
### Final Simplified Results
So the final simplified results are:
(i) [tex]\(\left(x^{-3}\right)^4 = \frac{1}{x^{12}}\)[/tex]
(ii) [tex]\(3 x^{\frac{1}{6}} \times 3 x^{-\frac{7}{6}} = \frac{9}{x}\)[/tex]
