Sure, let's work through each of these problems step-by-step:
### (i) [tex]\( 7^3 \times 7^4 \)[/tex]
Using the properties of exponents, specifically the rule that states when you multiply like bases, you add the exponents, we have:
[tex]\[ 7^3 \times 7^4 = 7^{3 + 4} = 7^7 \][/tex]
So, in exponential form, [tex]\( 7^3 \times 7^4 \)[/tex] simplifies to [tex]\( 7^7 \)[/tex].
### (ii) [tex]\( (-6) \times (-6)^5 \)[/tex]
Again, we apply the properties of exponents. Here, we notice that [tex]\( (-6) \)[/tex] can be expressed as [tex]\( (-6)^1 \)[/tex]:
[tex]\[ (-6) \times (-6)^5 = (-6)^1 \times (-6)^5 \][/tex]
Using the rule [tex]\( a^m \times a^n = a^{m+n} \)[/tex], we get:
[tex]\[ (-6)^1 \times (-6)^5 = (-6)^{1 + 5} = (-6)^6 \][/tex]
So, [tex]\( (-6) \times (-6)^5 \)[/tex] simplifies to [tex]\( (-6)^6 \)[/tex].
### (iii) [tex]\( 9^2 \times 9^{18} \times 9^7 \)[/tex]
Applying the same property of exponents, which states [tex]\( a^m \times a^n \times a^p = a^{m+n+p} \)[/tex], we get:
[tex]\[ 9^2 \times 9^{18} \times 9^7 = 9^{2 + 18 + 7} = 9^{27} \][/tex]
So, [tex]\( 9^2 \times 9^{18} \times 9^7 \)[/tex] simplifies to [tex]\( 9^{27} \)[/tex].
### Confirmation with Numerical Values:
For your reference, here are the numerical values of the results:
1. [tex]\( 7^7 \)[/tex] evaluates to 823543.
2. [tex]\( (-6)^6 \)[/tex] evaluates to 46656.
3. [tex]\( 9^{27} \)[/tex] evaluates to 58149737003040059690390169.
This confirms that our steps and usage of exponent properties are correct.
