Index Laws, Negative & Fractional Indices, and Base Transformations

Vocabulary practice flashcards covering basic, negative, and fractional index laws, radical conversions, base changes, and numerical rounding rules.

Index

A mathematical power or exponent that shows how many times a base number is multiplied by itself.

Indices

The plural form of index, referring collectively to powers or exponents.

Base Number

The primary number being repeatedly multiplied when raised to a power (for example, $5$ in the expression $5^4$).

Index Notation

A mathematical shortcut for writing repeated multiplications of the same number, such as expressing $5 \times 5 \times 5 \times 5$ as $5^4 = 625$.

First Law of Indices (Multiplication Law)

The rule stating that when multiplying powers with the same base, the exponents are added: $a^b \times a^c = a^{b+c}$.

Second Law of Indices (Division Law)

The rule stating that when dividing powers with the same base, the denominator exponent is subtracted from the numerator exponent: $\frac{a^b}{a^c} = a^{b-c}$.

Third Law of Indices (Power of a Power Law)

The rule stating that when raising a power to another power, the exponents are multiplied: $(a^b)^c = a^{b \times c}$.

Zero Index Law

The mathematical rule stating that any non-zero base raised to the power of zero equals one: $a^0 = 1$.

Negative Index Rule

The rule stating that a base raised to a negative exponent equals the reciprocal of the base raised to the positive exponent: $a^{-n} = \frac{1}{a^n}$.

Unit Fractional Index Rule

The rule stating that raising a base to the power of $\frac{1}{n}$ is equivalent to taking the $n\text{-th}$ root of the base: $a^{\frac{1}{n}} = \sqrt[n]{a}$.

General Fractional Index Rule

The rule stating that for a rational exponent $\frac{m}{n}$, the denominator represents the root and the numerator represents the power: $a^{\frac{m}{n}} = (\sqrt[n]{a})^m$.

Negative Fractional Index Rule

The rule combining the reciprocal and fractional root operations: $a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}} = \frac{1}{(\sqrt[n]{a})^m}$.

Reciprocal Rule for Fractional Bases

The rule stating that raising a fraction to a negative power inverts the fraction and changes the power to positive: $(\frac{a}{b})^{-n} = (\frac{b}{a})^n$.

Changing the Base

The algebraic technique of rewriting numerical bases as powers of a common, smaller base (such as expressing $16$ as $2^4$ or $27$ as $3^3$) to apply index laws.

Undefined Even Root of a Negative Number

An expression such as $(-9)^{\frac{1}{2}}$ or $(-16)^{\frac{1}{2}}$ that cannot be evaluated within the real number system because no real number multiplied by itself gives a negative result.

Surd

An irrational number expressed using a radical root sign ($\sqrt{\phantom{x}}$) that cannot be simplified into an exact fraction or terminating decimal.

Significant Figures (s.f.)

The meaningful digits in a number that contribute to its precision, counted starting from the first non-zero digit on the left.

Decimal Places (d.p.)

The number of digits located specifically to the right of the decimal point.

Nearest Integer Rounding

The process of rounding a decimal number to the closest whole number (for example, rounding $52.5917$ to $53$, or $0.1853$ to $0$).

Base-Exponent Multiplication Misconception

The common error of multiplying the base by the exponent instead of performing repeated multiplication (e.g., falsely writing $6^2 = 12$ instead of $36$, or $10^4 = 40$ instead of $10\,000$).

Tom's Calculator Rounding Mistake

The mistake of rounding the calculator value $516.158$ to $516.16$, which represents rounding to 2 decimal places rather than the requested 2 significant figures ($520$).

Rebecca's Consecutive Cubes Conjecture

The incorrect claim that adding three consecutive cube numbers always yields an odd answer; disproved because consecutive cubes alternate odd and even, producing $\text{odd} + \text{even} + \text{odd} = \text{even}$.

Tom's Cubing Misconception

The false assertion that cubing a number always makes it bigger, disproved by counterexamples such as $1^3 = 1$, $0^3 = 0$, or $(-2)^3 = -8 < -2$.

Power Pattern of Three

The descending division pattern showing that dividing successively by 3 defines zero and negative indices: $3^2 = 9$, $3^1 = 3$, $3^0 = 1$, $3^{-1} = \frac{1}{3}$, and $3^{-2} = \frac{1}{9}$.

Standard Form Rounding ($7.494 \times 10^7$)

The value $75\,000\,000$, obtained by rounding $7.494 \times 10^7$ to 2 significant figures and expressing the result as an ordinary number.

Evaluation of $(-2)^{-3}$

The calculation $\frac{1}{(-2)^3} = \frac{1}{-8} = -\frac{1}{8}$, demonstrating that an odd negative exponent on a negative base results in a negative fraction.

Evaluation of $5^{-2}$

The application of the negative exponent rule yielding $\frac{1}{5^2} = \frac{1}{25}$.

Negative Base with Even Exponent

The property that a negative base enclosed in brackets raised to an even power yields a positive value, such as $(-5)^2 = 25$ and $(-2)^2 = 4$.

Negative Base with Odd Exponent

The property that a negative base enclosed in brackets raised to an odd power preserves its negative sign, such as $(-8)^{\frac{1}{3}} = -2$ and $(-2)^3 = -8$.

Unbracketed Negative Power Expression

An expression like $-2^1 \times 2^1$, where the negative sign acts as a leading coefficient of $-1$, evaluating to $-(2 \times 2) = -4$.

Evaluation of $(4^{-2} \times 4^1)^2$

The calculation where exponents inside brackets add to $-1$ ($4^{-1}$), and multiplying by the outer exponent gives $4^{-2} = \frac{1}{16}$.

Evaluation of $5^{-1} + 5^{-1}$

An addition problem evaluated by converting terms to unit fractions: $\frac{1}{5} + \frac{1}{5} = \frac{2}{5}$.

Zero Exponent Product Misconception

The false belief that multiplying by a term with a zero exponent equals zero; since $10^0 = 1$, the correct evaluation is $10^3 \times 10^0 = 1000 \times 1 = 1000$.

Fraction-as-Power Misconception ($4^{-3}$)

The erroneous statement that $4^{-3} = \frac{3}{4}$, which confuses the negative index with a fraction of exponent over base; the true value is $\frac{1}{4^3} = \frac{1}{64}$.

Negative Power Inequality ($2^{-1} > 4^{-1}$)

The true mathematical inequality demonstrating that a smaller positive base produces a larger reciprocal value: $\frac{1}{2} > \frac{1}{4}$.

Mixed Number Fractional Index ($$(1\frac{7}{9})^{\frac{1}{2}}$ emerge)

The process of converting the mixed number into the improper fraction $\frac{16}{9}$ before taking the square root: $(\frac{16}{9})^{\frac{1}{2}} = \frac{4}{3}$.

Evaluation of $(\frac{8}{27})^{\frac{2}{3}}$

The evaluation found by taking the cube root of the fraction and then squaring the result: $(\frac{\sqrt[3]{8}}{\sqrt[3]{27}})^2 = (\frac{2}{3})^2 = \frac{4}{9}$.

Evaluation of $27^{-\frac{2}{3}}$

The calculation applying both negative reciprocal and root-power rules: $\frac{1}{(\sqrt[3]{27})^2} = \frac{1}{3^2} = \frac{1}{9}$.

Evaluation of $(\frac{4}{9})^{-\frac{1}{2}}$

The calculation involving inverting the fraction and taking the square root: $(\frac{9}{4})^{\frac{1}{2}} = \frac{\sqrt{9}}{\sqrt{4}} = \frac{3}{2}$.

Simplification of $(9x^2)^{\frac{1}{2}}$

Applying the power of $\frac{1}{2}$ to both the coefficient and the variable: $\sqrt{9} \times (x^2)^{\frac{1}{2}} = 3x$.

Power of an Algebraic Product ($(3a^2b^4)^3$)

Distributing the external power of 3 to every factor: $3^3 \times (a^2)^3 \times (b^4)^3 = 27a^6b^{12}$.

Simplification of $\frac{a^5 \times a^2}{a^{-3}}$

Combining indices by adding numerator powers and subtracting the denominator power: $a^{5+2 - (-3)} = a^{7+3} = a^{10}$.

Radical to Index Form Conversion

Expressing radical roots as fractional powers, such as writing $\sqrt[3]{x^2}$ as $x^{\frac{2}{3}}$ and $\sqrt{w^5}$ as $w^{\frac{5}{2}}$.

Single Power of 2 Form for $2\sqrt{2}$

The simplified expression obtained by writing $\sqrt{2}$ as $2^{\frac{1}{2}}$ and adding powers: $2^1 \times 2^{\frac{1}{2}} = 2^{\frac{3}{2}}$.

Exponential Equation Solving ($4^a = 16 \times 8^{2a}$)

Rewriting all terms in base 2 to form $2^{2a} = 2^4 \times 2^{6a}$, which gives $2a = 4 + 6a$ and yields $a = -1$.

Exponential Equation Solving ($9^x = (27^a)^{\frac{1}{2}} \times 3^b$)

Rewriting all terms in base 3 to form $3^{2x} = 3^{\frac{3}{2}a + b}$, which yields the expression $x = \frac{3}{4}a + \frac{1}{2}b$.

Ordering Fractional and Negative Powers

Evaluating terms into standard values to arrange them; for example, ordering $(-8)^{\frac{1}{3}} = -2$, $(\frac{8}{27})^{\frac{1}{3}} = \frac{2}{3}$, $4^{\frac{1}{2}} = 2$, and $16^{\frac{3}{4}} = 8$ from smallest to largest yields D, C, A, B.

Crack the Code: Harder Indices Total

The three-digit puzzle solution ($265$) obtained by calculating and summing all eighteen box values across the harder indices activity sheet.

Coefficient Division with Index Expressions

The process of dividing the numerical coefficients separately while subtracting matching base powers, such as $\frac{6y^7}{3y^2} = 2y^{7-2} = 2y^5$.

Index Equation Solving ($5^x \times 5^4 = 5^7$)

The method of equating exponents on like bases to solve for an unknown: $x + 4 = 7 \implies x = 3$.