How is half-life related to the base of the exponential function? How are indices used in logarithms for solving problems involving scales (e.g., Richter scale, pH scale)?
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what are indices in maths Informational
- What is the limit of $a^x$ as $x \to -\infty$ for $a > 1$? What is the limit of $a^x$ as $x \to \infty$ for $0 < a < 1$?
- The binomial theorem involves powers of a binomial expression. The coefficients in the binomial expansion are related to combinations. How are indices used in generating functions?
- How are indices used in combinatorics (e.g., counting principles)?
- Incorrectly applying the power of a power rule.
- Using the same definitions, $y/x = b^m / b^n = b^{m-n}$, so $\log_b (y/x) = m-n = \log_b y - \log_b x$. What are some examples of functions that involve indices in their definition?
- Simplify $\frac{x^3}{x^5}$.
- Simplify $(m^{-1})^{-3}$. Simplify $(3a^2b)^2$.
- Simplifying expressions with multiple nested exponents.
- Simplify $\sqrt{8}$ using indices. Simplify $\sqrt{12}$ using indices.
- Simplify $(p^{-2} q^{1/4})^{-8}$. Simplify $(a+b)^2$ (Expansion, not just index rules).
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