Indices, standard form and logarithms
An index tells how many times a base is multiplied by itself. In 2⁴, the base is 2 and the index is 4, giving 2 × 2 × 2 × 2 = 16. Indices provide a compact way to represent repeated multiplication. They are especially useful when working with very large measurements, small scientific quantities and expressions containing repeated algebraic factors.
When multiplying powers with the same base, add their indices: aᵐ × aⁿ = aᵐ⁺ⁿ. When dividing nonzero powers with the same base, subtract the indices. These laws do not allow unlike bases to be combined by simply adding their indices. For example, 2³ × 3² is 8 × 9, rather than 6⁵.
A raised to another multiplies the indices. A nonzero number to the zero equals one, while a negative index gives a reciprocal: a⁻ⁿ = 1/aⁿ. Thus 5⁻² is 1/25. The minus sign belongs to the index, so the result is not automatically a negative number. Keep brackets around a negative base when required.
Fractional indices connect powers to roots. For a positive number a, a^(1/2) means √a and a^(1/3) means its cube root. Also a^(m/n) can be evaluated by taking the nth root and then raising it to the m. For instance, 27^(2/3) = 3² = 9. Check the domain when working with even roots in real-number questions.
Standard form writes a positive number as A × 10ⁿ with 1 ≤ A < 10 and n an integer. The number 45,000 is 4.5 × 10⁴, while 0.00045 is 4.5 × 10⁻⁴. Count how far the decimal point moves and inspect the original size. A large positive number usually has a positive index; a small number between zero and one has a negative index.
A asks for an index: log₁₀1,000 = 3 because 10³ = 1,000. For positive arguments, logarithms turn multiplication into addition and division into subtraction. Do not apply these rules to addition inside an argument: log(a + b) is generally not log a + log b. In numerical work, show enough significant figures during intermediate steps and state any required rounding clearly.
Explain the laws and their restrictions
For a positive integer n, aⁿ means a multiplied by itself n times. When multiplying equal bases, aᵐ × aⁿ = aᵐ⁺ⁿ because the factors concatenate. Dividing equal nonzero bases cancels common factors, giving. Raising a to another multiplies the exponents. These are three different structures. In particular, 2³ + 2⁴ is. An addition sign does not allow the multiplication law.
For a ≠ 0, a⁰ = 1 because aᵐ/aᵐ equals both 1 and aᵐ⁻ᵐ. Also a⁻ⁿ = 1/aⁿ, so a negative indicates a reciprocal, not a negative answer. Thus. A fractional denotes a root: for positive a, and. Even roots of negative numbers are not real numbers. Keep track of brackets: (−3)² = 9, whereas −3² conventionally means −(3²) = −9.
Calculate with standard form
Standard form writes a nonzero number as A × 10ⁿ, where 1 ≤ |A| < 10 and n is an integer. For positive quantities this is usually stated as 1 ≤ A < 10. The number 0.00072 is 7.2 × 10⁻⁴; 6,300,000 is 6.3 × 10⁶. Standard form separates the significant digits from the scale. It is useful for populations, microscopic dimensions and scientific measurements, but it does not by itself tell you the precision of the original measurement.
To multiply, multiply the leading numbers and add the powers of ten. For example. To divide, divide the leading numbers and subtract exponents. For addition and subtraction, first express the terms with the same . Thus. Adding exponents here would be wrong.
Use logarithms as inverse operations
The statement logₐx = y means exactly aʸ = x, where a > 0, a ≠ 1 and x > 0. Hence log₂8 = 3 and log₁₀0.01 = −2. The can be negative even though its argument must be positive in real arithmetic. The rules, and logₐx follow from the index laws. There is no corresponding rule.
For an unfamiliar base, use logₐx = logᵦx/logᵦa, with a permitted base b such as 10. To solve 3ˣ = 20, take logarithms, so. Check that the answer lies between 2 and 3, since 9 < 20 < 27. In logarithmic equations, check each original argument after solving. From log₁₀(x − 1) = 1 we obtain x − 1 = 10, giving x = 11, which satisfies x > 1.
Simplify surds without losing exactness
A surd is an irrational root retained in exact form, such as √2. Factor out perfect squares. Like surds combine: cannot be combined into √5. Products can simplify: √6 × √24 = √144 = 12. The product rule for square roots is safe for nonnegative real factors; do not apply it indiscriminately to negative factors in real-number calculations.
Rationalising a denominator rewrites a fraction without a surd below the line. For 5/√3, multiply numerator and denominator by √3 to get 5√3/3. For 1/(2 + √3), multiply by the conjugate (2 − √3): the denominator becomes 4 − 3 = 1, giving 2 − √3. This works through the difference-of-squares identity. Self-check: simplify 16³ᐟ⁴, √50 + √8, and log₅125. Their answers are 8, 7√2, and 3. Keep exact forms when requested, and round only when the problem asks for an approximation.
Check a logarithmic candidate against its domain
Consider log₂(x − 1) + log₂(x + 1) = 3. Both arguments must be positive, so x > 1. Combine logs to obtain log₂(x² − 1) = 3, hence x² − 1 = 8 and. Only x = 3 meets the original domain. Squaring or combining expressions can create candidates that do not satisfy every original restriction. Checking just the final algebraic equation would incorrectly retain −3.
Also distinguish exact and approximate logarithms. The statement log₂8 = 3 is exact, while log₂7 ≈ 2.807 is rounded. If that rounded value is used in several later calculations, carry extra digits until the final answer. Estimate its interval first: because 4 < 7 < 8, the base-two must lie between 2 and 3.