To simplify the expression [tex]\(\omega^6 \cdot \omega^{-90}\)[/tex], follow these steps:
1. Identify the exponents: You have [tex]\(\omega^6\)[/tex] and [tex]\(\omega^{-90}\)[/tex]. Here, the base [tex]\(\omega\)[/tex] is the same in both terms, and the exponents are 6 and -90, respectively.
2. Use the properties of exponents: Recall the property of exponents which states that [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]. Applying this property, you add the exponents together since the base is the same.
3. Add the exponents:
[tex]\[
6 + (-90) = 6 - 90
\][/tex]
[tex]\[
6 - 90 = -84
\][/tex]
4. Combine the base with the new exponent: The new simplified expression will have the same base [tex]\(\omega\)[/tex] raised to the power of the summed exponents:
[tex]\[
\omega^{-84}
\][/tex]
Thus, the simplified form of [tex]\(\omega^6 \cdot \omega^{-90}\)[/tex] is [tex]\(\mathbf{\omega^{-84}}\)[/tex].
