Sure, let's simplify the given expression step-by-step. The expression to simplify is:
[tex]\[
\left[\frac{a^m}{a^n}\right]^p \left[\frac{a^n}{a^p}\right]^m \left[\frac{a^p}{a^m}\right]^n
\][/tex]
First, recall the property of exponents that [tex]\(\frac{a^x}{a^y} = a^{x-y}\)[/tex]. We can apply this property to each fraction in the expression:
1. Simplify [tex]\(\left[\frac{a^m}{a^n}\right]^p\)[/tex]:
[tex]\[
\left[\frac{a^m}{a^n}\right]^p = \left[a^{m-n}\right]^p = a^{p(m-n)}
\][/tex]
2. Simplify [tex]\(\left[\frac{a^n}{a^p}\right]^m\)[/tex]:
[tex]\[
\left[\frac{a^n}{a^p}\right]^m = \left[a^{n-p}\right]^m = a^{m(n-p)}
\][/tex]
3. Simplify [tex]\(\left[\frac{a^p}{a^m}\right]^n\)[/tex]:
[tex]\[
\left[\frac{a^p}{a^m}\right]^n = \left[a^{p-m}\right]^n = a^{n(p-m)}
\][/tex]
Now, we substitute these simplified forms back into the original expression:
[tex]\[
a^{p(m-n)} \cdot a^{m(n-p)} \cdot a^{n(p-m)}
\][/tex]
Next, we use the property of exponents that [tex]\(a^x \cdot a^y = a^{x+y}\)[/tex]. So, we combine the exponents of [tex]\(a\)[/tex]:
[tex]\[
a^{p(m-n) + m(n-p) + n(p-m)}
\][/tex]
Now, distribute the exponents:
[tex]\[
p(m-n) = pm - pn
\][/tex]
[tex]\[
m(n-p) = mn - mp
\][/tex]
[tex]\[
n(p-m) = np - nm
\][/tex]
Then combining these:
[tex]\[
pm - pn + mn - mp + np - nm
\][/tex]
Notice that some terms will cancel out. Combine like terms:
[tex]\[
pm - mp = 0
\][/tex]
[tex]\[
mn - nm = 0
\][/tex]
[tex]\[
np - pn = 0
\][/tex]
Therefore, the sum of the exponents will be:
[tex]\[
pm - pn + mn - mp + np - nm = 0
\][/tex]
So the simplified form of the original expression is:
[tex]\[
a^0 = 1
\][/tex]
Therefore, the given expression simplifies to:
[tex]\[
1
\][/tex]
This is the final simplified form.
