Learning Objectives
  • Use positive, zero, negative and fractional indices and apply index laws.
  • Solve simple exponential equations by expressing both sides with a common base.
  • Write, compare and calculate with numbers in standard form.
  • Simplify surds and rationalise denominators.
  • Use exponential multipliers to solve growth and decay problems.
Key Terms
Index
The power to which a number or expression is raised.
Base
The number or expression being raised to a power.
Fractional index
An index that represents a root and possibly a power.
Standard form
A number written as A\times10^n, where 1\le A<10.
Surd
An irrational root left in exact form.
Rationalise
Rewrite a fraction so that its denominator contains no surd.
Growth factor
The multiplier 1+r used for a fractional increase r.
Decay factor
The multiplier 1-r used for a fractional decrease r.
What This Chapter Covers
  • Index laws and fractional powers
  • Exponential equations with a common base
  • Standard form and calculations
  • Surd simplification and rationalisation
  • Compound growth and decay using multipliers
Index Laws

Index laws apply when bases are the same. They are consequences of repeated multiplication and division. The base must be non-zero when a zero or negative index is involved.

Law Meaning
a^m\times a^n=a^{m+n} Add indices when multiplying the same base.
a^m\div a^n=a^{m-n} Subtract indices when dividing the same base.
(a^m)^n=a^{mn} Multiply indices for a power of a power.
(ab)^n=a^nb^n Apply the power to every factor.
a^0=1 Any non-zero base to power zero equals 1.
a^{-n}=1/a^n A negative index gives a reciprocal.
a^{1/n}=\sqrt[n]{a} The denominator of a fractional index gives the root.
a^{m/n}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m The numerator gives the power.
Worked Example: Simplify Indices

Question: Simplify \frac{12x^5y^{-2}}{3x^{-1}y^3}, giving positive indices.

  1. Simplify the coefficient: 12/3=4.
  2. For x, subtract indices: 5-(-1)=6.
  3. For y, -2-3=-5.
  4. Rewrite y^{-5} as 1/y^5.

Answer: \frac{4x^6}{y^5}.

Worked Example: Fractional Index

Question: Evaluate 81^{3/4}.

  1. Take the fourth root first: 81^{1/4}=3.
  2. Cube the result: 3^3=27.

Answer: 27.

Solving Exponential Equations

When possible, express both sides as powers of the same base, then equate indices. Logarithms are not required for this syllabus.

Worked Example: Common Base

Question: Solve 3^{2x-1}=27.

  1. Write 27=3^3.
  2. Equate indices: 2x-1=3.
  3. 2x=4.

Answer: x=2.

Worked Example: Two Different-Looking Bases

Question: Solve 4^{x+1}=8^{x-1}.

  1. Write both sides in base 2: (2^2)^{x+1}=(2^3)^{x-1}.
  2. Equate indices: 2x+2=3x-3.

Answer: x=5.

Standard Form

A number is in standard form when it is written as A\times10^n with 1\le A<10 and integer n. Large numbers have positive powers of 10; small positive numbers have negative powers.

Worked Example: Converting To Standard Form

Question: Write 0.0000725 in standard form.

  1. Move the decimal point five places right to obtain 7.25.
  2. Moving right corresponds to a negative power.

Answer: 7.25\times10^{-5}.

Calculating In Standard Form

For multiplication, multiply the numerical parts and add powers of 10. For division, divide the numerical parts and subtract powers. For addition and subtraction, first rewrite both numbers with the same power of 10.

Worked Example: Multiplication

Question: Calculate (3.2\times10^5)(4.5\times10^{-3}) in standard form.

  1. Multiply: 3.2\times4.5=14.4.
  2. Add powers: 10^{5-3}=10^2.
  3. 14.4\times10^2=1.44\times10^3.

Answer: 1.44\times10^3.

Worked Example: Addition

Question: Calculate 6.3\times10^6+8.5\times10^5.

  1. Rewrite 8.5\times10^5=0.85\times10^6.
  2. Add coefficients: (6.3+0.85)\times10^6.

Answer: 7.15\times10^6.

Understanding Surds

A surd is an irrational root expressed exactly. Simplify by extracting perfect-square factors. Only like surds can be added or subtracted.

Worked Example: Simplifying Surds

Question: Simplify \sqrt{72}-\sqrt8+2\sqrt{18}.

  1. \sqrt{72}=\sqrt{36\times2}=6\sqrt2.
  2. \sqrt8=2\sqrt2.
  3. 2\sqrt{18}=2(3\sqrt2)=6\sqrt2.
  4. Combine like surds.

Answer: 10\sqrt2.

Multiplying Surds

Use \sqrt a\sqrt b=\sqrt{ab} for non-negative values. Expand brackets normally and simplify the result.

Worked Example: Surd Brackets

Question: Expand and simplify (3+\sqrt5)(2-\sqrt5).

  1. Expand: 6-3\sqrt5+2\sqrt5-5.
  2. Collect terms.

Answer: 1-\sqrt5.

Rationalising A Single-Surd Denominator

Multiply numerator and denominator by the surd in the denominator. The denominator then becomes rational.

Worked Example: One-Term Denominator

Question: Rationalise \frac{5}{\sqrt{10}}.

  1. Multiply by \frac{\sqrt{10}}{\sqrt{10}}.
  2. \frac{5\sqrt{10}}{10}.
  3. Simplify the numerical fraction.

Answer: \frac{\sqrt{10}}{2}.

Rationalising A Binomial Denominator

Use the conjugate. The product (a+b\sqrt c)(a-b\sqrt c) is a difference of squares and contains no surd term.

Worked Example: Conjugate

Question: Rationalise \frac{3}{2+\sqrt3}.

  1. Multiply numerator and denominator by 2-\sqrt3.
  2. Denominator: (2+\sqrt3)(2-\sqrt3)=4-3=1.

Answer: 6-3\sqrt3.

Exponential Growth And Decay

Repeated percentage change is multiplicative, not additive. A growth rate of r per period uses factor 1+r. A decay rate uses 1-r. Write percentages as decimals.

\text{Final amount}=\text{Initial amount}\times(1\pm r)^n
Worked Example: Compound Growth

Question: A population of 24 000 grows by 3.5% each year. Find the population after 6 years.

  1. Growth factor =1.035.
  2. Use 24000(1.035)^6.
  3. Evaluate on a calculator and round to a whole person.

Answer: Approximately 29 510.

Worked Example: Depreciation

Question: A machine worth 850 000 depreciates by 12% per year. Find its value after 4 years.

  1. Decay factor =1-0.12=0.88.
  2. Use 850000(0.88)^4.

Answer: Approximately 510 005, to the nearest unit.

Reverse Exponential Problems

If the final value is known, divide by the complete growth or decay factor. Do not simply reverse the percentage by adding or subtracting the same rate.

Worked Example: Find The Original Amount

Question: After 5 years of 4% annual growth, an investment is worth 121 665. Find the initial investment.

  1. 121665=P(1.04)^5.
  2. P=121665/(1.04)^5.

Answer: Approximately 100 000.

Examination Guidance
  • Use index laws only when the bases are the same.
  • Give final answers with positive indices unless a negative index is specifically requested.
  • Normalise standard-form answers so that the first number is at least 1 but less than 10.
  • Keep surds exact and simplify them fully before combining.
  • For compound change, use one multiplier raised to the number of periods and round only at the end.
Common Mistakes
  • Adding indices when adding powers, such as treating a^2+a^3 as a^5.
  • Using a^{-n}=-a^n instead of the reciprocal.
  • Writing 12.4\times10^5 as final standard form.
  • Combining unlike surds, such as \sqrt2+\sqrt3=\sqrt5.
  • Applying the percentage repeatedly by simple addition instead of compounding.
Knowledge Check And Practice

1. Simplify x^7\div x^{-2}.

Answer: x^9.

2. Evaluate 64^{2/3}.

Answer: 16.

3. Solve 5^{x+1}=125.

Answer: x=2.

4. Write 73 900 000 in standard form.

Answer: 7.39\times10^7.

5. Calculate (8\times10^9)\div(2\times10^3).

Answer: 4\times10^6.

6. Simplify \sqrt{50}+\sqrt8.

Answer: 7\sqrt2.

7. Rationalise \frac{4}{3-\sqrt5}.

Answer: 3+\sqrt5.

8. An amount decreases by 7% per year. State the annual multiplier.

Answer: 0.93.

9. A quantity of 500 grows by 8% for 3 periods. Write an exact calculator expression.

Answer: 500(1.08)^3.
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