91Ó°ÊÓ

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.

${\left(−\frac{a^m}{b^t}\right)}^{2t}={\left(\frac{−a^m}{b^t}\right)}^{2t}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰}\left[\text{±«²õ¾±²Ô²µâ€‰â¶Ä‰â¶Ä‰â¶Ä‰}−\frac{a}{b}=\frac{−a}{b}\text{ }\right]$

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.

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

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.

${\left(−a\right)}^{m×2t}=a^{2mt}$

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