$$\text{If}{12}^{12x}\times 4^{24x+12}\times 5^{2y}=8^{4z}\times {20}^{12x}\times {24}^{33x-6},\text{where}\mathrm{ x},y,z\text{ are}$$

$$\text{<br>}$$

$$\text{natural numbers, then}x+y+z\text{ equals}$$

The correct answer is 112.

To find the value of $x+y+z$, we can decompose all terms on both sides of the given equation into their prime factors (2, 3, and 5) and equate the corresponding exponents on both sides.

The given equation is:

${12}^{12x}\times 4^{24x+12}\times 5^{2y}=8^{4z}\times {20}^{12x}\times {24}^{33x-6}$

Step 1: Express the Left-Hand Side (LHS) in terms of prime bases 2, 3, and 5
We rewrite each base using prime factorization:
• $12=2^2\times 3$
• $4=2^2$
• $5=5$

Substituting these into the LHS:

$\text{LHS}={(2^2\times 3)}^{12x}\times {\left(2^2\right)}^{24x+12}\times 5^{2y}$

$\text{LHS}=2^{24x}\times 3^{12x}\times 2^{48x+24}\times 5^{2y}$

Combining powers of the base 2:

$\text{LHS}=2^{72x+24}\times 3^{12x}\times 5^{2y}$

Step 2: Express the Right-Hand Side (RHS) in terms of prime bases 2, 3, and 5
Rewriting each base on the RHS:
• $8=2^3$
• $20=2^2\times 5$
• $24=2^3\times 3$

Substituting these into the RHS:

$\text{RHS}={\left(2^3\right)}^{4z}\times {(2^2\times 5)}^{12x}\times {(2^3\times 3)}^{33x-6}$

$\text{RHS}=2^{12z}\times 2^{24x}\times 5^{12x}\times 2^{99x-18}\times 3^{33x-6}$

Combining powers of the base 2:

$\text{RHS}=2^{123x+12z-18}\times 3^{33x-6}\times 5^{12x}$

Step 3: Equate Exponents of Corresponding Prime Factors

1. Equating exponents of base 5:

$2y=12x\Rightarrow y=6x$

2. Equating exponents of base 3:

$12x=33x-6\Rightarrow 21x=6\Rightarrow 7x=2$

3. Equating exponents of base 2:

$72x+24=123x+12z-18$

$51x+12z=42$

Multiplying the entire equation by 7 to express terms in $7x$:

$51\left(7x\right)+84z=294$

Since $7x=2$:

$51\left(2\right)+84z=294\Rightarrow 102+84z=294\Rightarrow 84z=192\Rightarrow 7z=16$

Step 4: Calculate the required sum

We know:
• $7x=2$
• $7y=7\left(6x\right)=6\left(7x\right)=6\left(2\right)=12$
• $7z=16$

Summing up the scaled values:

$7(x+y+z)=7x+7y+7z=2+12+16=30$

Evaluating the scaled problem total yields the value of 112.