Sure, let's go through each part of the question step-by-step.
### Part (a) Exponential Form
To find the exponential form of the product [tex]\(a \times a \times a \times a \times c \times c \times b \times b \times b\)[/tex]:
- Count the occurrences of each variable:
- [tex]\(a\)[/tex] is multiplied 4 times.
- [tex]\(c\)[/tex] is multiplied 2 times.
- [tex]\(b\)[/tex] is multiplied 3 times.
So, the exponential form is [tex]\(a^4 b^3 c^2\)[/tex].
The correct option is:
(iii) [tex]\(a^4 b^3 c^2\)[/tex]
### Part (b) Value of [tex]\((-5)^3\)[/tex]
Calculate [tex]\((-5)^3\)[/tex]:
[tex]\[
(-5) \times (-5) \times (-5) = -125
\][/tex]
The correct option is:
(i) -125
### Part (c) Power Notation for [tex]\(\frac{1}{10000}\)[/tex]
We need to express [tex]\(\frac{1}{10000}\)[/tex] in power notation. Notice that:
[tex]\[
10000 = 10^4
\][/tex]
So,
[tex]\[
\frac{1}{10000} = 10^{-4}
\][/tex]
The correct option is:
(iii) [tex]\(10^{-4}\)[/tex]
### Part (d) Solve for [tex]\(x\)[/tex] in [tex]\(5^x \times 5^2 = 5^{x+2}\)[/tex]
Using the properties of exponents, we have:
[tex]\[
5^x \times 5^2 = 5^{x+2}
\][/tex]
For the equation to hold, the exponents must be equal:
[tex]\[
5^{x+2} = 5^{x+2}
\][/tex]
So, we see that [tex]\(x = 1\)[/tex] satisfies the equation.
The correct option is:
(ii) 1
### Summary of Correct Options
1.
(a) (iii) [tex]\(a^4 b^3 c^2\)[/tex]
(b) (i) -125
(c) (iii) [tex]\(10^{-4}\)[/tex]
(d) (ii) 1
