Solve (7×11×19)/(3×12×15) Raised to Power -4: Complex Fraction Challenge

Insert the corresponding expression:

$$$\left(\frac{7\times11\times19}{3\times12\times15}\right)^{-4}=$

The given expression is:
$\left(\frac{7\times11\times19}{3\times12\times15}\right)^{-4}$

To solve this expression, we need to apply the rules of exponents, specifically the rule for powers of a fraction. For any fraction$\left(\frac{a}{b}\right)^{-n}$, the expression is equivalent to$\left(\frac{b}{a}\right)^n$.
Therefore, negative exponents indicate that the fraction should be flipped and raised to the positive of that exponent.

Substitute the terms into this formula:
1. Flip the fraction: $\left(\frac{3\times12\times15}{7\times11\times19}\right)$
2. Raise both numerator and denominator to the power of 4:
Thus, we have:
$\left(\frac{3\times12\times15}{7\times11\times19}\right)^{4}$

Now evaluating each term individually:
- In the numerator:
- $3^4\times12^4\times15^4$
- In the denominator:
- $7^4\times11^4\times19^4$

Applying the negative exponent rule, each individual factor in both numerator and denominator should be inverted, altering the exponents to negative:
1. Numerator becomes: $3^{-4}\times12^{-4}\times15^{-4}$
2. Denominator becomes: $7^{-4}\times11^{-4}\times19^{-4}$

Rewriting the expression, we achieve:
$\frac{7^{-4}\times11^{-4}\times19^{-4}}{3^{-4}\times12^{-4}\times15^{-4}}$

This matches precisely the provided solution.

The solution to the question is:$\frac{7^{-4}\times11^{-4}\times19^{-4}}{3^{-4}\times12^{-4}\times15^{-4}}$