Power of a Product: Using multiple rules

$\left(8by\right)^{3\cdot y}+3^{x\cdot a}$

We begin by applying the following rule:

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

We then open the parentheses according to the above rule.

$\left(abc\right)^x=a^x\cdot b^x\cdot c^x$

$8^{3y}\cdot b^{3y}\cdot y^{3y}+3^{xa}$

In order to solve the problem we must use two power laws, as shown below:

A. Power property for terms with identical bases:

$\frac{a^m}{a^n}=a^{m-n}$B. Power property for an exponent raised to another exponent:

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

We will apply these two power laws to the problem in two steps:

Let's start by applying the power law specified in A to the second term from the left in the given problem:

$\frac{2^7}{2^4}=2^{7-4}=2^3$In the first step we apply the power law specified in A and then proceed to simplify the resulting expression,

We then advance to the next step and apply the power law specified in B to the third term from the left in the given problem :

$(8^2)^5=8^{2\cdot5}=8^{10}$In the first stage we apply the power law specified in B and then proceed to simplify the resulting expression,

Let's summarize the two steps listed above to solve the general problem:

$(4\cdot7)^9+\frac{2^7}{2^4}+(8^2)^5= (4\cdot7)^9+2^3+8^{10}$In the final step, we calculate the result of multiplying the terms within the parentheses in the first term from the left:

$(4\cdot7)^9+2^3+8^{10}=28^9+2^3+8^{10}$Therefore, the correct answer is option c.

In solving the problem, we use two laws of exponents, which we will mention:

a. The law of exponents for multiplying powers with the same 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 of exponents in solving the problem in two steps:

Let's start by applying the law of exponents mentioned in a' to the first expression on the left side of the problem:

$10^{-3}\cdot10^4=10^{-3+4}=10^1=10$When in the first step we applied the law of exponents mentioned in a' and in the following steps we simplified the expression that was obtained,

We continue to the next step and apply the law of exponents mentioned in b' and handle the third expression on the left side of the problem:

$(4^2)^5=4^{2\cdot5}=4^{10}$When in the first step we applied the law of exponents mentioned in b' and in the following steps we simplified the expression that was obtained,

We combine the two steps detailed above to the complete problem solution:

$10^{-3}\cdot10^4-(7\cdot9\cdot5)^3+(4^2)^5= 10-(7\cdot9\cdot5)^3+4^{10}$In the next step we calculate the result of multiplying the numbers inside the parentheses in the second expression on the left:

$10-(7\cdot9\cdot5)^3+4^{10}= 10-315^3+4^{10}$Therefore, the correct answer is answer b'.