What happens when the base is 1? Can the base be 0?
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what are indices in maths Informational
- Time and space complexity are often expressed using powers. What are some interesting mathematical problems or puzzles that involve indices? How can you prove the laws of indices using the definition of exponentiation?
- Rationalize $\frac{3}{\sqrt{5}}$.
- Simplify $x^{-4}$. What is a zero index?
- Solve for x and y $5^{x-y} = 25$ and $2^{x+y} = 16$. Solve for x and y $4^x = 8^y$ and $9^x = 3^{y+1}$.
- Simplify $\sqrt{\frac{a^4}{b^2}}$.
- Natural exponential function $e^x$ has important properties in calculus.
- Groups, rings, and fields have notions of powers of elements. How are indices related to the concept of dimension in mathematics and physics?
- Some production functions use Cobb-Douglas form with inputs raised to powers. How are indices used in environmental science (e.g., modeling population dynamics)? How are indices used in engineering (e.g., scaling laws)? How are indices used in computer graphics (e.g., transformations in 3D space)?
- What is the integral of $e^x$?
- Defining the base and the exponent based on the growth or decay rate. Fitting exponential curves to data. What are the limitations of using exponential models for real-world phenomena?
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