Number (Higher): Surds, Indices and Growth and Decay

Powers, roots and change over time

This unit gathers three higher number skills that look different but share one idea: repeated multiplication. Indices are a short way of writing repeated multiplication, so two to the power of five means two multiplied by itself five times. Surds are roots that do not come out as whole numbers, such as the square root of two, which you leave in exact form rather than rounding. Growth and decay use repeated percentage change, such as savings that grow each year or a car that loses value, worked out with a multiplier raised to a power. Master the laws of indices and the multiplier method, and all three become straightforward.

Number words for the higher tier

Learn these before you start. They name the powers, the roots and the repeated changes at the heart of this unit.

Match the index rule to its result

Growth against decay

Growth and decay are the same calculation with one difference: the size of the multiplier. Getting that right is the whole skill.

Which multiplier?

A price is to rise by 20 per cent. Which single multiplier increases an amount by 20 per cent?

• 1.2
• 0.2
• 0.8
• 20

Use a multiplier, not repeated steps

For repeated percentage change, do not add and take away the percentage again and again. Use one multiplier raised to a power. To increase by 10 per cent, multiply by 1.1; over three years, multiply by 1.1 to the power of three. To decrease by 10 per cent, multiply by 0.9. The multiplier method is faster, avoids mistakes, and is the only sensible way once there are many periods. Find the multiplier, raise it to the power of the number of periods, and multiply once.

Pick the true index facts

Select every statement about indices and surds that is true.

• Any non-zero number to the power of zero is one
• A negative power means one over the number to that power
• The power one half means the square root
• You add the powers when you divide
• A surd can always be written exactly as a whole number

Order a growth calculation

Put the steps of working out compound growth in order.

• Read the percentage change for one period
• Turn it into a multiplier, such as 1.1 for a rise of 10 per cent
• Raise the multiplier to the power of the number of periods
• Multiply the starting amount by this result
• Round the answer sensibly if it is needed
• Check it is bigger for growth or smaller for decay

Complete the higher number facts

A power that shows repeated multiplication is called an _____. A root that is not a whole number is a _____. The number you multiply by to make a percentage change is the _____. An amount that falls each year is an example of _____.

Savings over three years

Suppose you save 200 units and it grows by 5 per cent each year. Do not add 5 per cent three separate times. Instead, a rise of 5 per cent is a multiplier of 1.05. Over three years you multiply by 1.05 to the power of three. So the amount becomes 200 multiplied by 1.05 to the power of three, which works out at about 231 units. If instead the amount fell by 5 per cent each year, you would use 0.95 to the power of three and the amount would shrink. One multiplier, raised to the right power, handles the whole calculation at once.

Spot the correct working

Tap the TWO statements that show correct higher number working.

• To raise an amount by 10 per cent for two years, multiply by 1.1 to the power of two
• Any non-zero number to the power of zero equals one
• The square root of nine is a surd
• To decrease an amount by 20 per cent, multiply by 1.2

Work out the growth

A sum of 100 units grows by 10 per cent each year for two years. Using the multiplier 1.1 to the power of two, what is the sum worth after two years? Give your answer in units.

Choose the right method

For each problem, choose the correct method.

• You must increase an amount by 15 per cent each year for four years. What is the quickest correct method?
• You need to simplify the square root of twelve. What should you do?
• You must work out a non-zero number raised to the power of zero. What is the answer?

Build the multiplier rule

Choose the word or number for each gap to complete the rule for repeated percentage change.

Explain the multiplier method

A friend keeps adding and taking away percentages one year at a time and getting muddled. Explain the multiplier method for growth and decay, and why it is better.

• Explain what a multiplier is for a percentage change
• Explain the multiplier for growth and the multiplier for decay
• Explain how to handle several years at once
• Explain why this beats adding the percentage repeatedly
• Give one short worked example in words