Simplify: 4⁷·4⁻¹ + (3²)⁷ + 9⁵/9² | Exponent Operations

In solving the given problem we will use three laws of exponents, let's recall them:

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

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

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

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

c. The law of exponents for division of terms with identical bases:

$\frac{a^m}{a^n}=a^{m-n}$

We will apply these three laws of exponents to the expression in the problem in three stages:

Let's start by applying the law of exponents mentioned in a to the first term from the left in the expression:

$4^7\cdot4^{-1}= 4^{7+(-1)} = 4^{7-1}=4^{6}$

When in the first stage we applied the law of exponents mentioned in a and in the following stages we simplified the resulting expression,

We'll continue to the next stage and apply the law of exponents mentioned in b and deal with the second term from the left in the expression:

$(3^2)^7=3^{2\cdot7}=3^{14}$

When in the first stage we applied the law of exponents mentioned in b and in the following stages we simplified the resulting expression,

We'll continue to the next stage and apply the law of exponents mentioned in c and deal with the third term from the left in the expression:

$\frac{9^5}{9^2} =9^{5-2}=9^3$

When in the first stage we applied the law of exponents mentioned in c and in the following stages we simplified the resulting expression,

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

$4^7\cdot4^{-1}+(3^2)^7+\frac{9^5}{9^2} =4^6+3^{14}+9^3$

Therefore the correct answer is answer c.