Simplify: (9·7·6)³ + Powers of 9 + 7²⁵⁰ + 2⁴ Expression Challenge

When solving the following problem, we will use two laws of exponents:

a. The law of exponents for multiplying terms with identical bases:

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

b. The law of exponents for a power of a power:

$(a^m)^n=a^{m\cdot n}$

We will apply these two laws to the expression in the problem in two stages:

We'll start by applying the law of exponents mentioned in a to the second term from the left in the expression:

$9^{-3}\cdot9^4=9^{-3+4}=9^1=9$

Then, we'll apply the law of exponents mentioned in a in the first stage and simplified the resulting expression.

We'll continue to the next stage and apply the law of exponents mentioned in b and deal with the third term from the left in the expression. We'll do this in two steps:

$((7^2)^5)^6=(7^2)^{5\cdot6}=7^{2\cdot5\cdot6}=7^{60}$

In the first step we apply the law of exponents mentioned in b and eliminate the outer parentheses. In the next step, we apply the same law of exponents again and eliminate the remaining parentheses. In the following steps, we simplify the resulting expression.

Let's summarize the two stages detailed above for the complete solution of the problem:

$(9\cdot7\cdot6)^3+9^{-3}\cdot9^4+((7^2)^5)^6+2^4 = (9\cdot7\cdot6)^3+9+7^{60}+2^4$

In the next step we'll calculate the result of multiplying the terms inside the parentheses in the leftmost term:

$(9\cdot7\cdot6)^3+9+7^{60}+2^4 =378^3+9+7^{60}+2^4$

Therefore the correct answer is answer d.