Power Play – Formula & Equation Sheet
Essential formulas and equations from Ganita Prakash Part I, tailored for Class 8 in Mathematics.
This one-pager compiles key formulas and equations from the Power Play chapter of Ganita Prakash Part I. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formulas
Thickness after n folds: T = T₀ × 2ⁿ
T is the final thickness, T₀ is the initial thickness (0.001 cm), and n is the number of folds. This formula shows how the thickness doubles with each fold, illustrating exponential growth.
T₀ = 0.001 cm
T₀ is the initial thickness of the paper. This value acts as a baseline for calculating thickness after any number of folds.
Times increased by 10 folds: 2¹⁰ = 1024
This shows the multiplicative growth of the thickness of paper after 10 folds, indicating that the thickness increases by a factor of 1024 from the initial thickness.
Exponential growth: nᵃ
nᵃ represents n multiplied by itself a times. This general notation is applied to express how quantities increase rapidly, shown by examples like 2² or 5⁴.
√n = n¹/₂
This formula illustrates how to express square roots as fractional exponents, relevant in simplifying calculations involving powers.
Exponential notation: aᵇ × aᶜ = a⁽ᵇ+ᶜ⁾
Combining like bases in exponential expressions helps in simplifying multiplications, a fundamental arithmetic property of exponents.
aᵇ ÷ aᶜ = a⁽ᵇ−ᶜ⁾
This formula simplifies division involving exponents of the same base, which is critical in algebraic manipulations.
Volume of a cube: V = a³
V is the volume and a is the side length. Knowing this formula helps visualize exponential growth in three dimensions.
Volume of a cylinder: V = πr²h
Where r is the radius and h is the height. Understanding volume calculations in shapes links to concepts of growth in physical space.
For any positive integer n: n! = n × (n-1)!
This recursive definition of factorial relates to combinations and permutations, expanding the concept of growth into counting methods.
Equations
Thickness for 46 folds: T = 0.001 cm × 2⁴⁶
Calculating the thickness after 46 folds using the formula shows how quickly exponential growth leads to vast quantities, suitable for advanced problem-solving.
T(30) = 0.001 cm × 2³⁰ ≈ 10.7 km
Calculating thickness after 30 folds to demonstrate large-scale exponential growth visually, highlighting real-world implications.
Doubling rule: T(n) = 2 × T(n-1)
This recursive relationship aids in understanding how each fold affects the previous thickness.
If T(n) = T₀ × 2ⁿ, then n = log₂(T/T₀)
This logarithmic form allows the determination of the number of folds needed to achieve a certain thickness.
Total thickness after 10 folds: T(10) = 1.024 cm
Identifying the outcome after multiple folds gives context to exponential growth in a tangible way.
Estimated thickness after 20 folds: T(20) ≈ 10.4 m
Highlighting practical applications of exponential formulas by comparing estimated heights.
Thickness estimation after 27 folds: T(27) ≈ 1.3 km
A robust example demonstrating the scaling nature of exponential growth in calculated values.
Ratio of thickness after folds: R(n, m) = T(n)/T(m)
This ratio formula can be used to compare thickness at different folding points.
The effective increase after 3 folds: T(3) = 0.001 cm × 2³ = 0.008 cm
This equation summarizes the rapid increase in thickness as folds accumulate.
Comparative growth: G(n, m) = T(n) / T(m) = 2ⁿ⁻ᵐ
This equation showcases the comparative scaling factor of thickness between two different folding points.
