Power of a Product: Solving the equation

Recall the law of exponents for multiplication between terms with identical bases:

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

We'll use this to deal with the fraction's denominator in the problem:

$\frac{(-3)^5\cdot8^4}{(-3)^3(-3)^2(-3)^{-5}}=\frac{(-3)^5\cdot8^4}{(-3)^{3+2+(-5)}}=\frac{(-3)^5\cdot8^4}{(-3)^0}$

In the first stage, we'll apply the above law to the denominator and then proceed to simplify the expression with the exponent in the denominator.

Remember that raising any number to the power of 0 gives the result 1, or mathematically:

$X^0=1$

Therefore the denominator that we obtain in the last stage is 1.

This means that:

$\frac{(-3)^5\cdot8^4}{(-3)^3(-3)^2(-3)^{-5}}=\frac{(-3)^5\cdot8^4}{(-3)^0}=\frac{(-3)^5\cdot8^4}{1}=(-3)^5\cdot8^4$

Recall the law of exponents for an exponent of a product inside of parentheses is as follows:

$(x\cdot y)^n=x^n\cdot y^n$

Apply this law to the first term in the product:

$(-3)^5\cdot8^4=(-1\cdot3)^5\cdot8^4 =(-1)^5\cdot3^5\cdot8^4=-1\cdot 3^5\cdot 8^4=-3^5\cdot8^4$

Note that the exponent applies separately to both the number 3 and its sign, which is the minus sign that is in fact multiplication by $-1$.

Let's summarize everything we did:

$\frac{(-3)^5\cdot8^4}{(-3)^3(-3)^2(-3)^{-5}}=(-3)^5\cdot8^4 = -3^5\cdot8^4$

Therefore the correct answer is answer C.

To solve this problem, we'll follow these steps:

• Step 1: Evaluate the expression inside the parentheses $(3 \times 2 \times 4)$.
• Step 2: Square the result from Step 1.
• Step 3: Set the squared result equal to $3y$ and solve for $y$.

Now, let's work through each step:
Step 1: First, calculate $3 \times 2 = 6$.
Then multiply by 4: $6 \times 4 = 24$.
Step 2: Square the result: $24^2 = 576$.
Step 3: Set the equation $576 = 3y$.
To find $y$, divide both sides by 3:
$y = \frac{576}{3} = 192$.

Therefore, the solution to the problem is $y = 192$.