What is the difference between a power and a root?

A power multiplies a number by itself several times, for example \(3^2=9\). A root undoes it, for example \(\sqrt{9}=3\).

Why is any number to the power 0 equal to 1?

Because \(a^3\div a^3=a^{3-3}=a^0\) and any non-zero number divided by itself is 1.

When do I use a negative power in standard form?

Use a negative power for numbers smaller than 1, such as 0.0006 = 6 x 10^-4.

Year 8 powers, roots and standard form lesson to download as PDF

This Year 8 powers, roots and standard form lesson is written for pupils aged 12-13 and follows the National Curriculum. In short, it shows you how to use index laws for numbers, roots and standard form. Before you start, it helps to revise fractions, decimals and percentages, because this lesson builds on it.

Table of Contents

Year 8 powers, roots and standard form lesson: what you will learn

This lesson covers powers, roots, the index laws, prime factors and standard form. Learn the squares up to \(15^2\) and the cubes of 1 to 5 and 10 first.

Starter activity

Write down the number of seconds in a day. It is a big number: 86 400. Now write the number of seconds in a year. How many digits does your answer have? Why is it easier to write 3.15 \(\times 10^7\) than 31 536 000? Try the same for a very tiny thickness, the width of a spider's thread, which is about 0.000 003 m.

Powers and roots

Definition: power, base and index

A power (or index) tells you how many times to multiply a number by itself. In \(2^5\) the base is 2 and the index is 5, so \(2^5=2\times2\times2\times2\times2=32\). Recognise powers of 2 (2, 4, 8, 16, 32, 64, 128), of 3 (3, 9, 27, 81) and of 5 (5, 25, 125).

Definition: root

A root is the inverse of a power. The square root \(\sqrt{144}=12\) because \(12^2=144\). The cube root \(\sqrt[3]{27}=3\) because \(3^3=27\).

Worked example 1: estimating a root

Estimate \(\sqrt{50}\).

\(7^2=49\) and \(8^2=64\), so \(\sqrt{50}\) lies between 7 and 8, and it is very close to 7 (about 7.07).

The index laws

Rule: the index laws

For any non-zero base \(a\):

Multiplying: \(a^m\times a^n=a^{m+n}\). Example: \(2^3\times2^4=2^7=128\).
Dividing: \(a^m\div a^n=a^{m-n}\). Example: \(5^9\div5^6=5^3=125\).
Power of a power: \((a^m)^n=a^{mn}\). Example: \((3^2)^3=3^6=729\).
Zero index: \(a^0=1\). Example: \(7^0=1\).
Negative index: \(a^{-n}=\frac{1}{a^n}\). Example: \(2^{-3}=\frac{1}{8}\).

Key fact: why the laws work

Write \(2^3\times2^4\) out in full: \((2\times2\times2)\times(2\times2\times2\times2)\) has seven twos, so it equals \(2^7\). Write \(5^9\div5^6\) as a fraction and cancel six fives, leaving \(5^3\). The zero index follows too: \(a^3\div a^3=1\), and the law says \(a^{3-3}=a^0\), so \(a^0=1\).

Watch out: same base only

The multiplying and dividing laws only work when the bases are the same, and the base stays the same: \(2^3\times2^4=2^7\), not \(4^7\). A negative index does not make the answer negative: it means "reciprocal".

Worked example 2: using the laws

Simplify \(\dfrac{2^5\times2^{-2}}{2^2}\).

Top line: \(2^5\times2^{-2}=2^{5+(-2)}=2^3\). Divide: \(2^3\div2^2=2^{3-2}=2^1\). Answer: 2.

Worked example 3: a single power

Write \(\dfrac{3^7\times3^2}{3^4}\) as a single power of 3, then find its value.

Top: \(3^{7+2}=3^9\). Divide: \(3^{9-4}=3^5\). The value is \(3\times3\times3\times3\times3=243\).

Prime factors

Definition: prime factor

Every integer above 1 is a product of prime factors. Use powers to write the product neatly. A factor tree gives the same result however you start.

Worked example 4: prime factors of 72

Write 72 as a product of prime factors.

\(72=2\times36=2\times2\times18=2\times2\times2\times9=2^3\times3^2\).

Standard form

Definition: standard form

Standard form writes very large or very small numbers as \(A\times10^n\), where \(1\le A<10\) and \(n\) is an integer.

Large numbers have a positive \(n\): \(45\,000=4.5\times10^4\).
Small numbers have a negative \(n\): \(0.0006=6\times10^{-4}\).

Method: converting to and from standard form

Count how many places the decimal point moves to get a number \(A\) between 1 and 10.
Moving it left gives a positive power; moving it right gives a negative power.
To change back, move the point the number of places shown by \(n\): \(3.2\times10^5=320\,000\).

Worked example 5: multiplying and dividing

Work out \((2\times10^3)\times(3\times10^4)\) and \((8\times10^6)\div(2\times10^2)\).

Multiply the numbers: \(2\times3=6\). Add the powers: \(10^3\times10^4=10^7\). Answer: \(6\times10^7\). For the division, \(8\div2=4\) and \(10^6\div10^2=10^4\), so the answer is \(4\times10^4\).

Worked example 6: adjusting the answer

Work out \((4\times10^3)\times(5\times10^2)\).

\(4\times5=20\), so the product is \(20\times10^5\). But 20 is not between 1 and 10, so write \(20=2\times10^1\). The answer is \(2\times10^6\).

Worked example 7: adding

Work out \((6\times10^5)+(3\times10^4)\).

Change both numbers to the same power of 10 first: \(3\times10^4=0.3\times10^5\). Then \(6+0.3=6.3\), so the total is \(6.3\times10^5\).

Worked example 8: light from the Sun

Light travels at about \(3\times10^5\) km per second and the Sun is about \(1.5\times10^8\) km away. How long does light take to arrive?

Time \(=\) distance \(\div\) speed \(=\dfrac{1.5\times10^8}{3\times10^5}=0.5\times10^3=5\times10^2=500\) seconds, a little over 8 minutes.

Note: calculator tips

To enter \(3.2\times10^5\), type 3.2, press the exponent key (EXP or \(\times10^x\)), then 5. Do not type the multiplication sign and the 10 as well. A large number with a small raised number at the end in the display is standard form.

Watch out: common mistakes

Writing \(2^3\) as \(2\times3=6\) instead of \(2\times2\times2=8\).
Multiplying the bases when applying a law, such as \(2^3\times2^4=4^7\).
Leaving \(A\) outside the range 1 to 10, for example \(25\times10^3\).
Using a positive power for a number smaller than 1.
Forgetting that \(a^0=1\), not 0.

Worked example 9: roots and powers in context

A square patio has an area of 144 m². Find the side length, and then the length of fencing needed round it.

Side length \(=\sqrt{144}=12\) m. Perimeter \(=4\times12=48\) m. If the patio is a cube-shaped sandbox with volume 27 m³, the side is \(\sqrt[3]{27}=3\) m.

Worked example 10: negative indices

Work out \(\dfrac{2^{-1}\times2^{5}}{2^{3}}\).

Top: \(2^{-1+5}=2^4\). Divide: \(2^{4-3}=2^1=2\). Check with numbers: \(\dfrac12\times32=16\) and \(16\div8=2\).

Worked example 11: lowest common multiple with primes

Find the LCM of 24 and 36.

\(24=2^3\times3\) and \(36=2^2\times3^2\). For the LCM take the highest power of each prime: \(2^3\times3^2=8\times9=72\). For the HCF take the lowest power of each common prime: \(2^2\times3=12\).

Worked example 12: standard form with a calculator

Work out \((2.4\times10^{5})\times(5\times10^{-8})\).

Multiply the numbers: \(2.4\times5=12\). Add the powers of ten: \(10^{5+(-8)}=10^{-3}\). So the answer is \(12\times10^{-3}\), which is not in standard form. Move the decimal point: \(12=1.2\times10^1\), so the answer is \(1.2\times10^{-2}\). Check: \(2.4\times10^5=240\,000\) and \(5\times10^{-8}=0.00000005\), and \(240\,000\times0.00000005=0.012\).

Worked example 13: adding in standard form

Work out \(3.2\times10^{4}+5\times10^{3}\).

Write both with the same power of ten: \(5\times10^3=0.5\times10^4\). Then \(3.2\times10^4+0.5\times10^4=3.7\times10^4\). You cannot just add the numbers at the front when the powers are different.

Method: simplifying with the index laws

Check that every power has the same base. If not, try to rewrite them, for example \(4=2^2\) and \(8=2^3\).
Deal with the multiplying: add the indices on the top line.
Deal with the dividing: subtract the bottom index from the top index.
Write the answer as a single power, and then find its value if the question asks for a number.

Note: significant figures in standard form

When a question asks for an answer to 3 significant figures, the number in front of the power of ten must have 3 significant figures. For example, \(4.0375\times10^{13}\) becomes \(4.04\times10^{13}\). Keep every digit in your calculator while you work, and round only at the very end, because early rounding can change the last digit of your answer. Always write down the units, such as km, kg or seconds, and check that the size of your answer is sensible.

Watch out: the common traps

\(2^3\) means \(2\times2\times2=8\), not \(2\times3=6\).
\(2^3\times2^4=2^7\), but \(2^3+2^4\) cannot be simplified in this way.
\((-3)^2=9\), but \(-3^2=-9\) on a calculator.
A number in standard form must have \(1\le A<10\), so \(12\times10^3\) is not allowed.

Did you know?

In 1938 the American mathematician Edward Kasner asked his nine-year-old nephew to give a name to a very large number: 1 followed by 100 zeros. The boy called it a googol, written \(10^{100}\). The search company Google took its name, with a spelling change, from this idea. A googol is bigger than the number of atoms in the observable universe, which is about \(10^{80}\). Scientists need standard form because the universe contains numbers from the incredibly tiny, such as the mass of an electron, to the astonishingly huge, such as the distance across our galaxy, and both fit neatly on one line.

Why the index laws work

The index laws are not rules to memorise blindly: each one comes from writing the powers out in full. For example, \(2^{3}\times2^{4}\) means \((2\times2\times2)\times(2\times2\times2\times2)\), which is seven 2s multiplied together, so it equals \(2^{7}\). In the same way \(5^{6}\div5^{4}\) is six 5s on top and four 5s underneath; four pairs cancel and \(5^{2}\) is left. Following the same pattern down, \(5^{2}\div5^{2}=1\) shows why any number to the power 0 is 1, and \(5^{2}\div5^{3}=\dfrac{1}{5}\) shows why \(5^{-1}\) means one fifth.

Example: standard form with a calculator

Work out \((3.6\times10^{5})\div(4.5\times10^{-2})\).

Divide the numbers and the powers of 10 separately: \(3.6\div4.5=0.8\) and \(10^{5}\div10^{-2}=10^{5-(-2)}=10^{7}\). The result \(0.8\times10^{7}\) is not in standard form, because 0.8 is less than 1, so we write it as \(8\times10^{6}\). Always check that the first number is at least 1 and strictly less than 10 before you write your final answer.

Did you know?

A googol is \(10^{100}\), a 1 followed by a hundred zeros. The name was invented in 1920 by a nine-year-old boy, the nephew of an American mathematician, and the company Google later took its name from a misspelling of it.

Otto's tip

When the bases match, look at the powers only. Adding, subtracting or multiplying small whole numbers is much easier than working with giant values. Finish by checking that the first number in standard form is at least 1 and less than 10.

How to use this Year 8 powers, roots and standard form lesson

First, read the key points so that you know where the Year 8 powers, roots and standard form lesson is going. Then work through each definition and worked example, and cover the solution before you read it. After that, practise with the powers, roots and standard form worksheet and check every question with the worksheet answers. When you feel ready, test yourself with the multiple choice questions or the quick quiz. Finally, sit the powers, roots and standard form test and mark it with the test answers.

Also, the starters are a good warm-up, so you can keep the key facts fresh. Moreover, all the pages of this chapter work together, so you can move between them at any time. For example, many pupils read the lesson on Monday, practise on Wednesday and test themselves on Friday. In short, this Year 8 powers, roots and standard form lesson gives you everything you need for the chapter.

To sum up, regular practice works better than one long session. For instance, ten minutes a day is often enough. Besides, mistakes are useful, because they show what to revise next. Then, once a method feels easy, try a harder question. As a result, you build confidence step by step.