2.4 Powers, Square Roots, Powers of Ten, and Scientific Notation

Index Notation, Squares, Cubes, and Square Roots

Understanding Base and Index

Index notation (also called exponent or power notation) provides a compact way to write repeated multiplication of the same factor. In the expression $a^n$, the number $a$ is the base and the superscript $n$ is the index (or exponent), indicating how many times the base is multiplied by itself:

$$a^n = \underbrace{a \times a \times \cdots \times a}_{n\text{ factors}}$$

A classic non-calculator error is confusing $a^n$ with ordinary multiplication $a \times n$: for example, $2^4 = 2 \times 2 \times 2 \times 2 = 16$ (not $2 \times 4 = 8$), and $5^3 = 5 \times 5 \times 5 = 125$ (not $15$).

Benchmark Squares, Cubes, and Square Roots

For the non-calculator section and rapid geometry calculations (such as finding the side length of a square courtyard from its area $A = s^2$, or the volume of a cube $V = s^3$), you should recognise these benchmark squares, cubes, and square roots ($\sqrt{n}$, the positive number whose square equals $n$) instantly:

• Perfect Squares ($1^2$ to $15^2$): $1^2 = 1$, $2^2 = 4$, $3^2 = 9$, $4^2 = 16$, $5^2 = 25$, $6^2 = 36$, $7^2 = 49$, $8^2 = 64$, $9^2 = 81$, $10^2 = 100$, $11^2 = 121$, $12^2 = 144$, $13^2 = 169$, $14^2 = 196$, and $15^2 = 225$.
• Essential Cubes: $2^3 = 8$, $3^3 = 27$, $4^3 = 64$, $5^3 = 125$, and $10^3 = 1\,000$.
• Exact and Estimated Square Roots: Because $12^2 = 144$, $\sqrt{144} = 12$. To estimate a non-perfect square root such as $\sqrt{55}$ mentally, bracket $55$ between consecutive perfect squares: $49 < 55 < 64$, so $\sqrt{49} < \sqrt{55} < \sqrt{64}$, meaning $\sqrt{55}$ lies between $7$ and $8$ (slightly closer to $7$, around $7.4$).

Core Index Laws for Whole-Number Powers

The Four Foundational Index Laws

When expressions share the same base $a$ ($a \ne 0$), four foundational index laws simplify calculations without expanding every power:

1. Multiplying Powers with the Same Base (Add Indices): $a^m \times a^n = a^{m+n}$. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7 = 128$, and $10^2 \times 10^5 = 10^7$.
2. Dividing Powers with the Same Base (Subtract Indices): $a^m \div a^n = \frac{a^m}{a^n} = a^{m-n}$. For example, $5^6 \div 5^4 = 5^{6-4} = 5^2 = 25$, and $10^8 \div 10^5 = 10^3 = 1\,000$.
3. Power of a Power (Multiply Indices): $(a^m)^n = a^{m \times n}$. For example, $(3^2)^3 = 3^{2 \times 3} = 3^6 = 729$, and $(10^3)^2 = 10^6$.
4. The Zero Index Law: Any non-zero number raised to the power of zero equals $1$: $a^0 = 1$ (because $a^m \div a^m = a^{m-m} = a^0 = 1$). Thus $7^0 = 1$ and $10^0 = 1$.

Multiplying and Dividing by Powers of Ten

Decimal-Point Shifting and Metric Prefixes

Positive powers of ten ($10^n$) equal $1$ followed by $n$ zeros, whereas negative powers of ten ($10^{-n} = \frac{1}{10^n}$) represent decimal fractions with the digit $1$ in the $n\text{-th}$ decimal column:

• Multiplying by $10^n$ (or dividing by $10^{-n}$) shifts the decimal point $n$ places to the right, making the number larger (e.g., $3.405 \times 10^3 = 3\,405$).
• Dividing by $10^n$ (or multiplying by $10^{-n}$) shifts the decimal point $n$ places to the left, making the number smaller (e.g., $42.8 \times 10^{-2} = 42.8 \div 100 = 0.428$).

Scientific Notation (Standard Form)

Converting and Comparing Magnitudes

Science, geography, and education research stimuli frequently report very large or very small measurements in scientific notation (also called standard form):

$$a \times 10^n \quad \text{where } 1 \le a < 10 \text{ and } n \text{ is an integer}$$

The coefficient $a$ must have exactly one non-zero digit to the left of the decimal point ($1 \le a < 10$). Thus, $4.5 \times 10^6$ is valid scientific notation, whereas $45 \times 10^5$ and $0.45 \times 10^7$ equal $4\,500\,000$ but are not in strict standard form:

• Large Numbers ($n > 0$): A state education capital budget of $3,200,000 is written as $3.2 \times 10^6$ (the decimal point shifted $6$ places from the end to between $3$ and $2$).
• Small Numbers ($n < 0$): A bacterial cell width of $0.0000048\text{ m}$ is written as $4.8 \times 10^{-6}\text{ m}$ (the first non-zero digit $4$ sits in the sixth decimal place).

$$\frac{3.2 \times 10^6}{4.0 \times 10^5} = \frac{32 \times 10^5}{4.0 \times 10^5} = \frac{32}{4.0} = 8$$

Even though $3.2 < 4.0$, the number $3.2 \times 10^6$ ($3\,200\,000$) is 8 times larger than $4.0 \times 10^5$ ($400\,000$).