Simplify the Expression: a^y × a^x × 7^y × b^9 × a^6

First, we'll use the distributive property of multiplication and arrange the algebraic expression according to like bases:

$a^ya^xa^{6\text{ }}b^97^y$

Next, we'll use the power rule to multiply terms with the same base:

$a^m\cdot a^n=a^{m+n}$

Note that this rule applies to any number of terms in multiplication, not just two. For example, when multiplying three terms with the same base, we get:

$a^m\cdot a^n\cdot a^k=a^{m+n}\cdot a^k=a^{m+n+k}$

When we used the above power rule twice, we can perform the same calculation for four terms in multiplication, five and so on...

Therefore, we can combine all terms with the same base under one base:

$a^{y+x+6}b^97^y$

Note that we could only combine terms with identical bases using this rule,

From here we can see that the expression cannot be simplified further, and therefore this is the correct answer, which is answer C (since the distributive property of multiplication holds).