About Surds and Indices
A surd is a root that cannot be worked out exactly, and an index is a power. Both are manipulated by a small set of laws, and questions in this topic are really tests of whether those laws are known well enough to be applied without hesitation.
What you need to understand
- A power of a power multiplies the indices: a to the m, all raised to the n, is a to the mn.
- Multiplying powers of the same base adds the indices; dividing them subtracts the indices.
- Anything raised to the power zero is 1, and a negative index means the reciprocal.
- A surd is in its simplest form when the number inside has no square factor left.
- The square root of a product is the product of the square roots, which is what lets a surd be split.
- Two surds can be multiplied or divided directly, but they can be added only if the parts inside are identical.
Formulas to remember
How to work through these questions
- Deal with the indices first and the numbers afterwards, because the index laws are mechanical.
- When more than one law applies, work from the innermost bracket outwards.
- To simplify a surd, look for the largest square factor of the number inside it.
- To solve an equation with the unknown in the index, write both sides as powers of the same base and then equate the indices.
- Check by substituting a small numerical case, which catches a misapplied law immediately.
Mistakes that cost marks
- Adding the indices when the powers are being multiplied inside a bracket instead of multiplying them.
- Assuming the square root of a sum equals the sum of the square roots.
- Leaving a surd unsimplified when the question asks for its simplest form.
- Taking the square outside the root instead of its square root, so that root 48 becomes 48 root 2.
- Substituting the number for the index, which turns a power into an ordinary product.
Worked example
Simplify the square root of 448.
- Look for the largest square factor of 448. Since 448 = 64 x 7 and 64 is a perfect square, that factor is 64.
- So the square root of 448 is the square root of 64 multiplied by the square root of 7.
- The square root of 64 is 8, which gives 8 root 7.
- Check: 8 root 7 squared is 64 x 7 = 448, so the simplification is correct.
Answer: 8 root 7
Practice questions with answers
A few Surds and Indices questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
Find the value of $(2^{3})^{2} \div 2^{4}$.
-
A
16777216
-
B
4
-
C
1024
-
D
48
Answer: Option B — with explanation
A power of a power multiplies the indices, and division subtracts them.
$(2^{3})^{2} = 2^{6}$, and $2^{6} \div 2^{4} = 2^{2} = 4$.
Common mistakes
- subtracted the two values instead of the indices: 48
- multiplied the indices again at the division step: 16777216
- added the indices instead of subtracting them: 1024
Question 2
Solve for x: $5^{x + 3} = 625$.
Answer: Option B — with explanation
Write the right hand side as a power of 5: $625 = 5^{4}$.
So $x + 3 = 4$, giving $x = 4 - 3 = 1$.
Common mistakes
- divided by the constant instead of subtracting it: 1.33
- added the constant instead of subtracting it: 7
- read off the index of the right hand side and stopped: 4
Question 3
Express the surd $\sqrt{486}$ in simplest form.
-
A
$81\sqrt{6}$
-
B
$6\sqrt{9}$
-
C
$\sqrt{486}$
-
D
$9\sqrt{6}$
Answer: Option D — with explanation
Look for the largest square factor of 486: $486 = 81 \times 6$.
So $\sqrt{486} = \sqrt{81} \times \sqrt{6} = 9\sqrt{6}$.
Common mistakes
- swapped the numbers inside and outside the surd: $6\sqrt{9}$
- left the surd untouched: $\sqrt{486}$
- took the square outside instead of its square root: $81\sqrt{6}$
Question 4
Simplify $(2^{3})^{2} \div 2^{4}$.
-
A
1024
-
B
16777216
-
C
48
-
D
4
Answer: Option D — with explanation
A power of a power multiplies the indices, and division subtracts them.
$(2^{3})^{2} = 2^{6}$, and $2^{6} \div 2^{4} = 2^{2} = 4$.
Common mistakes
- subtracted the two values instead of the indices: 48
- multiplied the indices again at the division step: 16777216
- added the indices instead of subtracting them: 1024
Question 5
Find x if $2^{x + 2} = 32$.
Answer: Option C — with explanation
Write the right hand side as a power of 2: $32 = 2^{5}$.
So $x + 2 = 5$, giving $x = 5 - 2 = 3$.
Common mistakes
- read off the index of the right hand side and stopped: 5
- added the constant instead of subtracting it: 7
- divided by the constant instead of subtracting it: 2.5
Frequently asked questions
What is a surd?
A surd is a root of a number that cannot be written exactly as a fraction or a terminating decimal. The square root of 2 is a surd, while the square root of 9 is not, because it is exactly 3.
How do I know when a surd is in its simplest form?
When the number inside the root has no square factor left except 1. Root 12 is not simplified, because 12 has the square factor 4 and it becomes 2 root 3.
Can I add two surds?
Only if the numbers inside the roots are the same. Root 3 plus root 3 is 2 root 3, but root 3 plus root 5 cannot be combined.
What does a negative index mean?
It means the reciprocal of the positive power. Two to the power minus three is one over two cubed, which is one eighth.
