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:

x1^x2^x3^x4^x5^x6^x7^x8^x
112345678
21491625364964
3182764125216343512
411681256625129624014096
51322431024312577761680732768
616472940961562546656117649262144
71128218716384781252799368235432097152
81256656165536390625279936478296916777216

Would you like more details about how the powers of each base grow as x increases?

Here are five related questions for further exploration:

  1. How do the powers of each base change as x increases?
  2. What is the behavior of exponential growth in general?
  3. How do we calculate higher powers of numbers efficiently (e.g., using logarithms)?
  4. What happens when x is negative for different bases?
  5. How do powers and exponents relate to logarithms?

Tip: When working with powers and exponents, always remember the basic exponent rules like am×an=am+na^m \times a^n = a^{m+n}, and (am)n=am×n(a^m)^n = a^{m \times n}, 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