Alright, let's find the reciprocals of the given expressions step-by-step.
### (i) [tex]\( 6^0 \times 9^0 \)[/tex]
First, we evaluate each power:
- [tex]\( 6^0 \)[/tex]: Any non-zero number raised to the power of 0 is 1. Therefore, [tex]\( 6^0 = 1 \)[/tex].
- [tex]\( 9^0 \)[/tex]: Similarly, [tex]\( 9^0 = 1 \)[/tex].
Now, multiply these results together:
[tex]\[ 6^0 \times 9^0 = 1 \times 1 = 1 \][/tex]
Next, we need the reciprocal of 1. The reciprocal of a number [tex]\( x \)[/tex] is [tex]\(\frac{1}{x} \)[/tex]. So, the reciprocal of 1 is:
[tex]\[ \frac{1}{1} = 1.0 \][/tex]
### (ii) [tex]\(\left( - \frac{5}{7} \right)^7 \)[/tex]
First, we determine the value of [tex]\(\left( - \frac{5}{7} \right)^7 \)[/tex]. This is:
[tex]\[ \left( - \frac{5}{7} \right)^7 \][/tex]
The value remains an expression which typically could be a fraction, simplifying exponentially isn't straightforward here.
We now find the reciprocal of this expression. Reciprocal means taking [tex]\( \frac{1}{x} \)[/tex].
Thus, the reciprocal of [tex]\(\left( - \frac{5}{7} \right)^7 \)[/tex] is:
[tex]\[ \frac{1}{\left( - \frac{5}{7} \right)^7} \approx -10.541350399999999 \][/tex]
### (iii) [tex]\((-4)^3\)[/tex]
First, we compute the value of [tex]\((-4)^3 \)[/tex]. This is:
[tex]\[ (-4)^3 = -4 \times -4 \times -4 = -64 \][/tex]
Now, we need to find the reciprocal of -64. The reciprocal of a number [tex]\( x \)[/tex] is [tex]\(\frac{1}{x} \)[/tex].
So, the reciprocal of [tex]\((-64)\)[/tex] is:
[tex]\[ \frac{1}{-64} = -0.015625 \][/tex]
### Summary
Here are the reciprocals of the given expressions:
1. [tex]\( 6^0 \times 9^0 \)[/tex]: The reciprocal is [tex]\( 1.0 \)[/tex].
2. [tex]\(\left( - \frac{5}{7} \right)^7\)[/tex]: The reciprocal is approximately [tex]\( -10.541350399999999 \)[/tex].
3. [tex]\( (-4)^3 \)[/tex]: The reciprocal is [tex]\( -0.015625 \)[/tex].
These are the detailed step-by-step solutions to find the reciprocals of each expression.
