Math Problem Statement
complete this "powers table" x 1^x 2^x 3^x 4^x 5^x 6^x 7^x 8^x 1 1 2 2 1 4 3 1 8 4 1 16 5 1 32 6 1 64 7 1 128 8 1 256
Solution
Here’s the completed "powers table" with all the values filled in:
x | 1^x | 2^x | 3^x | 4^x | 5^x | 6^x | 7^x | 8^x |
---|---|---|---|---|---|---|---|---|
1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
2 | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 |
3 | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 |
4 | 1 | 16 | 81 | 256 | 625 | 1296 | 2401 | 4096 |
5 | 1 | 32 | 243 | 1024 | 3125 | 7776 | 16807 | 32768 |
6 | 1 | 64 | 729 | 4096 | 15625 | 46656 | 117649 | 262144 |
7 | 1 | 128 | 2187 | 16384 | 78125 | 279936 | 823543 | 2097152 |
8 | 1 | 256 | 6561 | 65536 | 390625 | 279936 | 4782969 | 16777216 |
Would you like more details about how the powers of each base grow as x increases?
Here are five related questions for further exploration:
- How do the powers of each base change as x increases?
- What is the behavior of exponential growth in general?
- How do we calculate higher powers of numbers efficiently (e.g., using logarithms)?
- What happens when x is negative for different bases?
- How do powers and exponents relate to logarithms?
Tip: When working with powers and exponents, always remember the basic exponent rules like , and , which can simplify complex calculations.
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Math Problem Analysis
Mathematical Concepts
Exponents
Powers of Numbers
Exponential Growth
Formulas
a^x = a * a * ... * a (x times)
Theorems
Exponentiation Rule
Suitable Grade Level
Grades 6-8