Indices and Standard Form

Indices and Standard Form

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Indices (exponents) are the small superscript numbers that tell you how many times a base multiplies itself: in a^n, a is the base and n is the index (also called the exponent or power). The six laws of indices are the only manipulation tools you need:

• a^m × a^n = a^(m+n)
• a^m ÷ a^n = a^(m−n)
• (a^m)^n = a^(mn)
• a^0 = 1 (provided a ≠ 0)
• a^(−n) = 1 / a^n
• a^(m/n) = ⁿ√(a^m) (so a^(1/n) = ⁿ√a)

Standard form (scientific notation) writes any number as N = a × 10^n where 1 ≤ a < 10 and n is an integer. Moving the decimal right decreases n; moving it left increases n.

NCEE hot spots: simplifying surds via fractional indices, evaluating 2^(−3), 27^(2/3), and writing quantities like 0.000034 as 3.4 × 10^(−5).

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🟡 Standard — Regular Study

Standard content for students with a few days to months.

The Language of Indices

In the expression a^n, a is the base and n is the index (or exponent/power). Read a^n as “a to the power n”. The notation 5^3 means 5 × 5 × 5 = 125. An index of 1 is usually omitted (a^1 = a), and a^0 = 1 for any non-zero base. Indices let us write very large or very small products compactly.

The Six Laws of Indices

When a and b are non-zero:

Law	Rule	Use it for
Multiplication	a^m × a^n = a^(m+n)	Same base, multiplied
Division	a^m ÷ a^n = a^(m−n)	Same base, divided
Power of a power	(a^m)^n = a^(mn)	Bracketed exponent
Zero index	a^0 = 1 (a ≠ 0)	Any non-zero base to 0
Negative index	a^(−n) = 1 / a^n	Reciprocal form
Fractional index	a^(m/n) = ⁿ√(a^m)	Roots and powers combined

For example, 2^(−3) = 1/2^3 = 1/8, and 27^(2/3) = (27^(1/3))^2 = 3^2 = 9.

Standard Form (Scientific Notation)

Any number can be written as N = a × 10^n with 1 ≤ a < 10 and n ∈ ℤ. The coefficient a has exactly one non-zero digit before the decimal point.

Converting in: count how many places the decimal moves from its original position to sit just after the first non-zero digit. Each place moved right makes n negative; each place moved left makes n positive.

NCEE Question Patterns

Paper 1 (objective) tests direct evaluation and simplification, while Paper 2 (theory) asks 2–5 mark questions on standard-form conversions and arithmetic with numbers in standard form. Always re-check that your coefficient a lies in [1, 10).

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🔴 Extended — Deep Study

Comprehensive coverage for students on a longer study timeline.

Edge Cases and Power Distribution

Two extra rules matter when bases differ:

• (ab)^n = a^n × b^n
• (a/b)^n = a^n / b^n (with b ≠ 0)

Watch for the trap (ab)^n ≠ a^n × b — the exponent distributes across both factors. Likewise, (a/b)^n ≠ a/b^n.

A subtle but high-yield point: 0^0 is undefined, not 1. Any answer claiming 0^0 = 1 loses the mark.

Combining Fractional and Integer Indices

To evaluate 8^(2/3), apply the fractional index first (the n-th root), then the integer index: 8^(2/3) = (8^(1/3))^2 = 2^2 = 4. Equally valid: 8^(2/3) = (8²)^(1/3) = 64^(1/3) = 4. The two routes agree because indices satisfy (a^m)^n = a^(mn).

Standard-Form Arithmetic

When multiplying (a × 10^m) × (b × 10^n), multiply a × b first, then add the powers of ten; if the resulting coefficient falls outside [1, 10), adjust the power. For division, divide coefficients and subtract powers. For addition/subtraction, first rewrite both numbers with the same power of ten so the coefficients can be added/subtracted directly.

Common Mistakes

• Adding bases: writing 2^3 × 2^4 as 4^7 (should be 2^7).
• Sign slip on negatives: 2^(−3) written as −8.
• Coefficient out of range: 56.3 × 10^4 (correct: 5.63 × 10^5).
• Distributing exponents incorrectly: (ab)^n done as a^n × b.

Practice Prompts

1. Simplify (3x²)³ ÷ (9x⁴) and express the result with positive indices.
2. Write 0.000407 in standard form, then compute (4.07 × 10^(−4)) × (5 × 10^6), giving the answer back in standard form.

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