For any nonzero real numbers a and b and any integers m and t, simplify the expression${\left(-\frac{a^m}{b^t}\right)}^{2t}$and describe each step.

When two variables with different bases, but same indices are divided, we are required to divide the bases and raise the same index to it.

${\left(\frac{a}{b}\right)}^p=\frac{a^p}{b^p}$

When a variable with some index is again raised with different index, then both the indices are multiplied together raised to the power of the same base.

${\left(a^p\right)}^q=a^{pq}$

Rewrite the given expression.

${\left(-\frac{a^m}{b^t}\right)}^{2t}={\left(\frac{-a^m}{b^t}\right)}^{2t}\left[\text{Using\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}-\frac{a}{b}=\frac{-a}{b}\right]$

Use the above two rules to simplify the given expression.

Apply rule 7 to the expression ${\left(\frac{-a^m}{b^t}\right)}^{2t}$.

${\left(\frac{-a^m}{b^t}\right)}^{2t}=\frac{{\left(-a^m\right)}^{2t}}{{\left(b^t\right)}^{2t}}$

Now apply rule 5 to the expression width="78" height="52" role="math">−am2tbt2t.

$\begin{array}{c}\frac{{\left(-a^m\right)}^{2t}}{{\left(b^t\right)}^{2t}}=\frac{{\left(-a\right)}^{m\times 2t}}{{\left(b\right)}^{t\times 2t}}\\ \frac{{\left(-a^m\right)}^{2t}}{{\left(b^t\right)}^{2t}}=\frac{{\left(-a\right)}^{2mt}}{{\left(b\right)}^{2t^2}}\end{array}$

Since the power of ‘a’ is 2mt, and 2mt will always be an even number as it’s a multiple of 2.

Apply the exponent rule ${\left(-a\right)}^n=a^n$, if n is even.

${\left(-a\right)}^{m\times 2t}=a^{2mt}$

Therefore, the simplified expression is as follows:

$\begin{array}{l}\frac{{\left(-a^m\right)}^{2t}}{{\left(b^t\right)}^{2t}}=\frac{{\left(a\right)}^{2mt}}{{\left(b\right)}^{2t^2}}\\ \frac{{\left(-a^m\right)}^{2t}}{{\left(b^t\right)}^{2t}}=\frac{a^{2mt}}{b^{2t^2}}\end{array}$

Therefore ${\left(-\frac{a^m}{b^t}\right)}^{2t}=\frac{a^{2mt}}{b^{2t^2}}$.