Indices and Surds

Prerequisites: exact fractions, prime factorisation and expanding brackets. The diagram uses Pythagoras’ theorem: the square of a right triangle’s hypotenuse equals the sum of the squares of its two legs. All practice questions in this lesson are original, not official TMUA questions.

What This Lesson Is About

Indices and surds are two ways of writing powers and roots without replacing them by rounded decimals. This lesson explains what these symbols mean, which rules preserve their values, and how to use them safely in calculations and equations.

We begin with powers, including negative and fractional indices. We then distinguish a square-root value from the solutions of an equation, simplify root expressions, and remove roots from denominators. Finally, we examine why squaring an equation can produce answers that need to be rejected.

1. Indices: Expressing Repeated Multiplication and Roots

An index, also called an exponent, tells you how a base is being raised to a power. Understanding its meaning first makes the laws of indices easier to use and helps explain why zero, negative and fractional indices need care.

1.1 Positive powers and the laws of indices

For a positive integer exponent, a power represents repeated multiplication: 2³ = 2 × 2 × 2. Multiplying 2³ by 2² joins three copies of 2 to two more, giving 2⁵. The index laws extend this consistent pattern to a wider range of powers.

For a positive real base a, the rules a^m × a^n = a^(m + n), a^m/a^n = a^(m − n), and (a^m)^n = a^(mn) apply to rational exponents m and n. The caret means “raised to the power”; parentheses group the entire exponent. A quotient must have a nonzero denominator.

1.2 Zero and negative indices: the identity and reciprocal

The definitions a^0 = 1 and a^(−n) = 1/a^n fit these rules for a ≠ 0. A negative exponent does not make a positive base negative: 2^(−3) = 1/8. The expression 0^0 is not assigned a value in these lessons.

A reciprocal is the number that multiplies the original nonzero number to give 1. For example, 8 × (1/8) = 1. Thinking of a negative index as a reciprocal avoids confusing 2^(−3) with −2³.

1.3 Fractional indices: roots written as powers

A root reverses an integer power: a fifth root of 32 is a number whose fifth power is 32. Fractional exponents provide notation for taking such roots and then raising them to further powers.

For positive a, a^(1/q) is the positive qth root and a^(p/q) = (a^(1/q))^p, where p is an integer and q is a positive integer. Negative bases need separate care: odd roots such as the cube root of −8 are real, but the square root of −8 is not real. Do not apply nested rational-power rules to negative bases without checking the definitions.

2. Square Roots: Principal Values and Equation Solutions

Fractional indices introduced roots as values. Before simplifying them, we need to separate the value selected by the square-root symbol from all the numbers that solve a squared equation.

2.1 The principal square root: one nonnegative value

The symbol √u denotes the nonnegative square root of u, and is real only for u ≥ 0. Therefore √9 = 3, while the equation x² = 9 has two solutions, x = 3 and x = −3. The root symbol and the equation answer different questions.

2.2 The modulus connection: why √(x²) equals |x|

Squaring hides the sign of a real number: 4 and −4 both square to 16. Taking the principal square root cannot recover that lost sign, so it returns the number’s nonnegative magnitude instead.

For a real number x, √(x²) = |x|. The modulus |x| is x when x ≥ 0 and −x when x < 0, so it is always nonnegative. At x = −4 the expression √(x²) equals 4, not −4.

2.3 An exact geometric length: the unit-square diagonal

A root does not have to describe an abstract calculation. The next diagram shows how √2 arises as a measurable length whose exact value is most naturally written with a root symbol.

A unit square gives a useful exact length. Triangle ABC has perpendicular sides AB = BC = 1, so AC² = 1² + 1² = 2. A length is positive, hence AC = √2. Writing √2 preserves this length exactly; 1.41 would be only an approximation.

Figure 1. Pythagoras gives AC² = 2. The positive diagonal length is exactly √2.

3. Simplifying Surds: Extracting and Combining Factors

A surd is an irrational root expression, such as √2 or √3. Simplifying it means finding a more useful exact form, not rounding its decimal expansion. We first extract square factors, then combine terms with the same remaining root.

3.1 Extracting square factors from a radicand

The radicand is the expression inside the root symbol. Look for a perfect-square factor inside it, because the square root of that factor can be evaluated exactly.

For nonnegative a and b, √(ab) = √a √b. Extract perfect-square factors: √72 = √(36 × 2) = 6√2. The same rule does not apply to sums. For example, √(9 + 16) = 5, but √9 + √16 = 7.

3.2 Like surds: adding coefficients of the same root

Combine surds only after identifying the same irrational factor, just as you combine like terms in algebra. Thus 2√3 + 5√3 = 7√3, while √2 + √3 cannot be combined into one multiple by adding the radicands.

In 2√3 + 5√3, both terms count multiples of the same quantity √3, so only their coefficients are added. Roots with different remaining radicands are different quantities and cannot be combined in this way.

4. Rationalising a Denominator

A fraction containing a root in its denominator is already a valid exact number when that denominator is nonzero. Rationalising rewrites it with a rational denominator, which can make comparison or further calculation more convenient.

4.1 A single square root: multiply by the same root

To rationalise a denominator, multiply numerator and denominator by the same nonzero expression so the denominator becomes rational. For a single root, 1/√a = √a/a when a > 0.

4.2 A two-term denominator: use the conjugate

For a denominator involving a sum or difference, multiplying by just one root usually leaves another root term. Instead, use a paired expression chosen so that the cross-products cancel.

The conjugate of u + v is u − v. Their product is u² − v² because the middle terms cancel. When v is a square root, this difference can remove the root from the denominator. The original denominator and the conjugate used as a multiplier must be nonzero.

5. Root Equations: Why Squaring Needs a Check

The earlier sections changed the form of numbers. We now use roots in equations, where the task is to find allowed values of a variable. Removing a root by squaring is useful, but it can weaken the original requirement.

5.1 Squaring preserves equality but can hide a sign difference

For example, 3² = (−3)² is true even though 3 = −3 is false. A candidate satisfying the squared equation may therefore fail the original equation. Such an extra candidate is called an extraneous root.

If two real expressions are equal, their squares are equal. The reverse implication can fail because opposite numbers have the same square. Before squaring a root equation, note that the principal root is nonnegative; afterwards substitute every candidate into the original equation.

6. Practice

These questions follow the lesson sequence: indices, surd simplification, rationalising, sign restrictions and root equations. The final comparison asks you to choose a method that keeps both values exact. State any sign condition your reasoning uses.

1. Evaluate 81^(3/4) and 16^(−3/2).

2. Simplify √50 + √8 − √18.

3. Rationalise 2/(√3 + 1).

4. If x < 0, simplify √(9x²).

5. Solve √(2x + 3) = x for real x.

6. Which is larger, √7 + √2 or √15? Give an exact comparison.

Worked Solutions

1. Interpret fractional and negative exponents

81^(1/4) = 3, so 81^(3/4) = 27. Also √16 = 4, so 16^(−3/2) = 4^(−3) = 1/64. Check that 27⁴ = 81³ and 64² = 16³.

2. Extract square factors before combining

5√2 + 2√2 − 3√2 = 4√2. Squaring each extracted term recovers its original radicand.

3. Multiply by a conjugate

Multiply by (√3 − 1)/(√3 − 1): 2(√3 − 1)/(3 − 1) = √3 − 1. Check: (√3 − 1)(√3 + 1) = 2.

4. Use the sign restriction in √(x²)

√(9x²) = 3|x| = −3x because x < 0. At x = −2, both sides equal 6, illustrating why 3x would have the wrong sign.

5. Reject the extraneous root

Any solution has x ≥ 0. Squaring gives x² − 2x − 3 = 0, or (x − 3)(x + 1) = 0. The candidate 3 works since √9 = 3; −1 fails since √1 ≠ −1. Only x = 3.

6. Compare positive quantities by their squares

Both quantities are positive, so compare squares. (√7 + √2)² = 9 + 2√14 and (√15)² = 15. Since √14 > 3, we have 9 + 2√14 > 15. Therefore √7 + √2 is larger. Positivity is what makes comparing squares sufficient.

7. Summary

• Use exponent laws with their base restrictions.
• The principal square root is nonnegative, so √(x²) = |x|.
• Extract square factors and use conjugates to keep expressions exact.
• Squaring can add false candidates; check the original equation.