Hard GCSE Integer indices Questions

Understand the goal

We need the first power of $3$ that passes $1000$, so build up the powers of $3$ until one goes over.

Start the list

Begin with $3$ to the power $1$.

Next power

Multiply by $3$: nine.

Next power

Multiply by $3$ again: twenty-seven.

Next power

$27$ times $3$ is $81$.

Next power

$81$ times $3$ is $243$. Still well below $1000$.

Next power

$243$ times $3$ is $729$ — close to $1000$, but not past it yet.

Compare with $1000$

$3$ to the power $6$ falls short, so $n = 6$ does not satisfy the inequality.

Next power

$729$ times $3$ is $2187$.

Compare with $1000$

$3$ to the power $7$ is past $1000$ — the inequality is satisfied for the first time.

Bracket the answer

$1000$ is trapped between the sixth and seventh powers of $3$, so the smallest winning index is $7$.

Check the multiplication $729 \times 3$

Checking the key step: $700 \times 3 = 2100$ and $29 \times 3 = 87$, giving $2187$. ✓

Why no smaller n works

Powers of $3$ grow every time n goes up, so every index below $7$ gives an even smaller value than $729$, all below $1000$.

Watch for the common error

It is tempting to stop at $729$ because it is close, but the question needs strictly greater than $1000$.

State the answer

The smallest integer n with $3$ to the power n greater than $1000$ is $7$.

Plan the simplification

Simplify the two brackets first, then combine the top, then divide. Every index law will get used.

Recall the power-of-a-power law

A power of a power means multiplying the indices.

Simplify the top bracket

$2$ times $3$ gives $6$.

Rewrite the numerator

The top of the fraction is now a product of two powers of x.

Recall the product law

Multiplying powers of the same base means adding the indices.

Combine the numerator

$6$ plus $4$ gives $10$, so the whole top is x to the power $10$.

Simplify the bottom bracket

$5$ times $2$ gives $10$, so the bottom is also x to the power $10$.

Rewrite the fraction

The fraction is a power divided by an identical power.

Recall the quotient law

Dividing powers of the same base means subtracting the indices.

Subtract the indices

$10$ minus $10$ gives the zero index.

Recall the zero index rule

Any non-zero value to the power $0$ equals $1$.

State the simplified value

The expression simplifies completely to the number $1$ — the x disappears.

Check with a value: set $x = 2$

Test with $x = 2$: the top is $64$ times $16$ and the bottom is $1024$.

Work out the check

$64$ times $16$ is $1024$, so the fraction is $1024$ over $1024$, which is $1$. ✓

State the answer

The fully simplified value is $1$ (for any non-zero x).

Understand the goal

Working out $2$ to the power $40$ exactly by hand would take ages. The given fact lets us estimate it quickly using the index laws.

State the given fact

$2$ to the power $10$ is $1024$, which is very close to $1000$ — that is, $10$ cubed.

Connect $40$ to $10$

The index $40$ is four lots of $10$, which suggests grouping the power in tens.

Recall the power-of-a-power law

Reading the law in reverse lets us group $2$ to the power $40$ as a power of $2$ to the power $10$.

Group the power

$2$ to the power $40$ is $2$ to the power $10$, raised to the power $4$.

Substitute the approximation

Replace $2$ to the power $10$ with roughly $10$ cubed.

Apply the law again

A power of a power means multiplying the indices: $3$ times $4$.

Simplify the index

So $2$ to the power $40$ is approximately $10$ to the power $12$.

Interpret the size

$10$ to the power $12$ is a $1$ followed by $12$ zeros — one trillion.

Judge the direction of the error

Because $1024$ is slightly more than $1000$, the true value is slightly bigger than our estimate.

Quantify the error roughly

The small factor of $1.024$ gets raised to the power $4$, giving roughly a $10$% underestimate — still an excellent approximation.

Compare with the exact value

The exact value is about $1.1$ trillion, confirming the estimate and the direction of the error.

Reflect on the technique

This trick converts powers of $2$ into powers of $10$, which is exactly how computer scientists relate kilobytes $(2^10)$ to thousands $(10^3)$.

Watch for the common error

Do not multiply $1024$ by $4$ — raising to a power means repeated multiplication of the whole power, not multiplying the index into the value.

State the answer

$2$ to the power $40$ is approximately $10$ to the power $12$, about one trillion.

Look at the first equation

The bases $2$ and $8$ are different, so write $8$ as a power of $2$ to compare the indices.

Write $8$ as a power of $2$

$8$ is $2$ cubed.

Substitute into the first equation

Replace $8$ with $2$ cubed, keeping the index b outside.

Recall the power-of-a-power law

A power of a power means multiplying the indices.

Simplify the right side

$3$ times b gives $3$b in the index.

Equate the indices

Equal powers of $2$ must have equal indices, so a is three times b.

Bring in the second fact

The question also says the two indices add up to $12$. We now have two simple equations.

Substitute $a = 3b$

Replace a with $3$b in the sum, so only one unknown remains.

Collect the like terms

$3$b plus b makes $4$b.

Solve for b

Dividing both sides by $4$ gives b equals $3$.

Find a

Multiply b by $3$: a is $9$.

Check the sum

The indices add to $12$, as required. ✓

Check the powers

$2$ to the power $9$ is $512$, and $8$ cubed is also $512$, so the first equation holds too. ✓

Reflect on the method

Changing to a common base turned a power equation into easy simultaneous equations — a very common exam technique.

State the answer

The values are a equals $9$ and b equals $3$.

Look at the structure of the equation

The bases $9$ and $27$ are different, but both are powers of $3$, so convert everything to base $3$.

Write $9$ as a power of $3$

$9$ is $3$ squared.

Write $27$ as a power of $3$

$27$ is $3$ cubed.

Substitute into the equation

Replace $9$ and $27$ with their base-$3$ forms.

Recall the power-of-a-power law

A power of a power means multiplying the indices on each side.

Simplify the left side

$2$ times x gives $2$x.

Simplify the right side

$3$ times the whole bracket $x - 1$. Keep the brackets until they are expanded.

Expand the bracket in the index

Multiply out: $3$ times x and $3$ times negative $1$.

Rewrite the equation

Both sides are now single powers of $3$.

Equate the indices

Equal powers of $3$ must have equal indices, giving a linear equation with x on both sides.

Subtract $2$x from both sides

Gather the x terms on one side, as in the equations topic.

Solve for x

Adding $3$ to both sides gives x equals $3$.

Check the left side

$9$ cubed is $729$.

Check the right side

$27$ squared is also $729$, so both sides match. ✓

State the answer

The solution of the equation is x equals $3$.