A-Level Indices and surds Practice Questions

Recall the multiplication law

When you multiply powers that share the same base you keep the base and add the indices. Here the base is $x$ in both parts. This is one of the basic laws of indices.

Add the indices

The two indices are 5 and 3, so we add them. The base $x$ does not change.

Work out the sum

Since $5+3=8$, the answer is $x$ to the power 8. That is already in its simplest form.

Simplify the first surd

Before adding surds they must look the same. Simplify $\sqrt8$ using its square factor 4.

Rewrite the sum

Now both terms contain $\sqrt2$, so they are like surds.

Add the like surds

$2\sqrt2+1\sqrt2=3\sqrt2$, treating $\sqrt2$ like a common factor.

Same base means subtract indices

Both top and bottom have base 2, so we use the division law even though the indices contain a letter.

Apply the division law

Subtract the whole bottom index from the top index. Using brackets stops sign mistakes.

Remove the brackets

The minus sign flips the sign of both terms in $(n-1)$.

Simplify the index

The $n$ terms cancel, leaving 2.

Write the result

So the expression equals $2^2=4$, independent of $n$.

Check with a value of n

Testing $n=1$ gives $2^2/2^0=4$, matching our answer. This confirms the result really does not depend on $n$.

Notice the equal indices

Both powers have the same index $x$ but different bases. There is a law that lets us join them.

Combine the left side

When two powers share the index, multiply the bases: $a^n b^n=(ab)^n$.

Simplify the base

$2\times3=6$, so the left side is $6^x$.

Rewrite the equation

The equation is now a single power equal to 36.

Write 36 as a power of 6

Since $6^2=36$, express the right side with base 6.

Match the bases

Both sides are now powers of 6.

Solve

Equating the indices gives $x=2$.

Recall the law we used

The key move was joining two powers with the same index into a single power of the product.

Restate the tidied equation

Both sides ended up as single powers of 6, which made the comparison of indices possible.

Check the solution

Substituting $x=2$: $4\times9=36$, which matches. So the answer is confirmed.

Plan the approach

Both surds are nested. If each equals a perfect square, the outer roots simplify to simple surd expressions.

Try a square for the first surd

Test $(\sqrt5+2)^2$ using $(p+q)^2$.

Confirm the first square

The numbers give $5+4=9$, which matches.

Take the first square root

Since $\sqrt5+2>0$, this is the correct root.

Try a square for the second surd

Test $(\sqrt5-2)^2$ using $(p-q)^2$.

Confirm the second square

The numbers give $5+4=9$, which matches.

Check the sign of √5 − 2

Since $\sqrt5$ is bigger than 2, $\sqrt5-2$ is positive, so it is the correct root.

Take the second square root

The positive root is $\sqrt5-2$.

Set up the subtraction

Now carry out the subtraction from the original expression.

Remove the brackets

The minus sign flips the signs of the second bracket.

Simplify

The $\sqrt5$ terms cancel and $2+2=4$.

Recall the expansion identities

Recognising perfect squares of the form $(\sqrt5\pm2)^2$ was the key idea for de-nesting these surds.

Note both roots are positive

Both expressions under the outer roots are positive, so each square root is a positive surd expression.

Explain why the answer is whole

Because the two roots differ only in the sign of the constant, subtracting them cancels the surds and leaves a whole number.

Check numerically

The two roots are about 4.236 and 0.236, and their difference is 4, confirming the answer.