To compare [tex]\((-8.4)^5\)[/tex] and [tex]\(2^2\)[/tex] without evaluating their exact values, consider the following points:
1. Magnitude of the Base Numbers:
- The absolute value of [tex]\(-8.4\)[/tex] is [tex]\(8.4\)[/tex], which is significantly larger than the base [tex]\(2\)[/tex].
- In an exponentiation operation, the larger the base number, the larger the result, assuming positive exponents.
2. Sign of the Result for Negative Base:
- [tex]\(-8.4\)[/tex] is a negative number. When raised to an odd power (like [tex]\(5\)[/tex]), the result remains negative.
- [tex]\(2^2\)[/tex] is a positive number since any positive number raised to any power will remain positive.
3. General Behavior of Exponents:
- For positive exponents, the magnitude (absolute value) of [tex]\((-8.4)^5\)[/tex] will be [tex]\(8.4\)[/tex] raised to the power of [tex]\(5\)[/tex], which is a very large number.
- [tex]\(2^2\)[/tex] evaluates to [tex]\(4\)[/tex], which is a much smaller number compared to the magnitude of [tex]\(8.4^5\)[/tex].
Conclusion:
Given that [tex]\((-8.4)^5\)[/tex] is negative and a large number while [tex]\(2^2\)[/tex] is a small positive number, it is clear that [tex]\((-8.4)^5\)[/tex] is less than [tex]\(2^2\)[/tex]. This is because any negative number (no matter its magnitude) will always be less than a positive number.
