Sure, let's work on each part of the problem step-by-step.
### Part (i):
Simplify [tex]\(\left(\frac{6}{7}\right)^{-1}\)[/tex]:
When you raise a fraction to the power of -1, you take its reciprocal. The reciprocal of [tex]\(\frac{6}{7}\)[/tex] is [tex]\(\frac{7}{6}\)[/tex].
So, [tex]\[\left(\frac{6}{7}\right)^{-1} = \frac{7}{6}\][/tex]
### Part (iii):
Simplify [tex]\(\frac{\left(2^5\right)^3 \times 7^3}{8^3 \times 7}\)[/tex]:
1. Firstly simplify the numerator: [tex]\((2^5)^3\)[/tex] and [tex]\(7^3\)[/tex]:
[tex]\[ (2^5)^3 = 2^{5 \times 3} = 2^{15} \][/tex]
[tex]\[ 7^3 = 7^3 \][/tex]
2. Simplify the denominator [tex]\(8^3 \times 7\)[/tex]:
[tex]\[ 8^3 = (2^3)^3 = 2^{3 \times 3} = 2^9 \][/tex]
[tex]\[ 8^3 \times 7 = 2^9 \times 7 \][/tex]
3. Now, put them into the fraction:
[tex]\[ \frac{2^{15} \times 7^3}{2^9 \times 7} \][/tex]
4. Simplify the powers of 2 and 7:
[tex]\[ = \frac{2^{15-9} \times 7^{3-1}}{1} \][/tex]
[tex]\[ = 2^6 \times 7^2 \][/tex]
[tex]\[ = 64 \times 49 \][/tex]
5. Calculate the final value:
[tex]\[ 64 \times 49 = 3136 \][/tex]
So, [tex]\[\frac{\left(2^5\right)^3 \times 7^3}{8^3 \times 7} = 3136\][/tex]
### Part (iv):
Simplify [tex]\(\left[\left\{\left(-\frac{1}{3}\right)^2\right\}^{-2}\right]^{-1}\)[/tex]:
1. Compute [tex]\(\left(-\frac{1}{3}\right)^2\)[/tex]:
[tex]\[ \left(-\frac{1}{3}\right)^2 = \frac{1}{9} \][/tex]
2. Then, raise [tex]\(\frac{1}{9}\)[/tex] to the power of -2:
[tex]\[ \left(\frac{1}{9}\right)^{-2} = 9^2 \][/tex]
[tex]\[ 9^2 = 81 \][/tex]
3. Now, since we need to raise it to the power of -1:
[tex]\[ (81)^{-1} = \frac{1}{81} \][/tex]
However, [tex]\(\left[\left\{\left(-\frac{1}{3}\right)^2\right\}^{-2}\right]^{-1} = 81\)[/tex].
So, [tex]\[\left[\left\{\left(-\frac{1}{3}\right)^2\right\}^{-2}\right]^{-1} = 81\][/tex]
### Summary of Results:
(i) [tex]\(\left(\frac{6}{7}\right)^{-1} = \frac{7}{6} = 1.1666666666666667\)[/tex]
(iii) [tex]\(\frac{\left(2^5\right)^3 \times 7^3}{8^3 \times 7} = 3136\)[/tex]
(iv) [tex]\(\left[\left\{\left(-\frac{1}{3}\right)^2\right\}^{-2}\right]^{-1} = 81\)[/tex]
These are the simplified results for each part of the problem.
