Animated Solution for Mathematics - Basic Mathematics: The product of all positive real values of $x$ satisfying the equation $x^{(16({\log }_{5}x{)}^3-68{\log }_{5}x)}=5^{-16}$ is ______.

We are presented with the equation:

This problem features a variable in both the base and the exponent, combined with logarithmic terms. To simplify this, we apply the logarithmic lens by taking ${\log }_5$ on both sides.

Applying ${\log }_5$ to both sides and utilizing the power rule ${\log }_a(m^n)=n\cdot {\log }_{a}m$, we bring the exponent down:

Using the property ${\log }_a(a^k)=k$, the right-hand side simplifies to $-16$. Our equation now becomes:

To manage the complexity, we introduce the substitution $t={\log }_{5}x$. This transforms the equation into a polynomial:

Distributing $t$ yields the bi-quadratic equation:

Letting $u=t^2$, we simplify the expression to $16u^2-68u+16=0$. Dividing the entire equation by $4$, we obtain:

We factorize the quadratic by splitting the middle term:

This yields two possible values for $u$: $u=\frac{1}{4}$ or $u=4$. Since $u=t^2$, we solve for $t$: 1. If $t^2=\frac{1}{4}$, then $t=\pm \frac{1}{2}$. 2. If $t^2=4$, then $t=\pm 2$.

Translating back to $x$ using $x=5^t$, the four possible values for $x$ are $5^{1/2}$, $5^{-1/2}$, $5^2$, and $5^{-2}$. The product of these values is:

The final product of all possible values of $x$ is 1.