Why does 8 to the power of minus 5 equal a positive number and not minus 8 to the power of 5?

Because the minus sign is in the index, not in front of the base. A negative index means one over: a to the minus n equals 1 divided by a to the n. So 8 to the minus 5 equals 1 divided by 8 to the 5, which equals 1 divided by 32768. That is a small positive fraction. Minus 8 to the 5 is minus 32768, a large negative integer.

What is the reciprocal of 8 to the power 5, and how do you write it as a single power?

The reciprocal of 8 to the power 5 is 1 divided by 8 to the power 5. To write that as a single power, keep the base 8 and negate the index: 8 to the minus 5. The traps to avoid are minus 8 to the 5 (a sign change, not a reciprocal) and 5 to the minus 8 (swapping the base and index).

How do you tell whether an expression with a minus sign gives a positive or a negative result?

Check where the minus sign sits. If the minus is in the index (like 8 to the minus 5), the result is a positive reciprocal. If the minus is attached to the base and the power is odd (like negative 2 all cubed), the result is negative. These are two different structures.

GCSE Maths Higher · AQA · Index laws and standard form

Negative indices mean reciprocals, not negative numbers

When students see $8^{-5}$ , the instinct is to read the minus in the index as a sign on the result and write $-8^5$ . That answer is wrong in sign and wrong in magnitude. The correct rule is $a^{-n} = \frac{1}{a^n}$ : a negative index means one over, not a sign change.

So $8^{-5} = \frac{1}{8^5} = \frac{1}{32768}$ , a tiny positive fraction. The trap answer $-8^5 = -32768$ is a large negative integer. These differ in sign and in magnitude by a factor of $32768^2$ . This confusion is one of the most reliable grade 7 to 9 mark-losers on AQA Higher index-law papers.

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How to spot it in your own work

You wrote $-8^5$ (or any negative number) as the value of $8^{-5}$ , reading the minus in the index as a sign on the result.
You circled $-8^5$ , $5^{-8}$ , or $5^8$ as the reciprocal of $8^5$ instead of $8^{-5}$ .
Given $7^n = x$ , you wrote $7^{-n} = -x$ instead of $7^{-n} = \frac{1}{x}$ .
You confused a negative index ( $8^{-5}$ , positive result) with a negative base to an odd power ( $(-2)^3 = -8$ , negative result).

An exam question that triggers it

Here is the structure of a typical AQA Higher index-laws question:

Circle the expression that is equal to the reciprocal of $8^5$ .

$8^{-5}$ $-8^5$ $5^{-8}$ $5^8$

The misconception circles $-8^5$ (sign change) or $5^{-8}$ (base-index swap). The fix: the reciprocal of $8^5$ is $\frac{1}{8^5} = 8^{-5}$ . Keep the base 8 and negate only the index.

Why students fall for this

Students see a minus sign and associate it with “negative number.” The minus in $8^{-5}$ is in the index, but the brain registers the sign first and attaches it to the result: the expression becomes $-8^5$ in the student’s mental model.

A second error pattern is the base-index swap: the student sees the 5 in the index of $8^5$ and the 8 in the base and produces $5^{-8}$ or $5^8$ as the “reciprocal,” conflating reciprocal with some kind of reversal operation on the digits.

Both errors are reinforced by the fact that the wrong answers look plausible: they use the same digits as the correct expression. A student who writes $-8^5$ or $5^{-8}$ often has no sense that anything is wrong, which is what makes this a reliable mark-loser at grades 7 to 9.

The fix: Negative index means one over: a^-n = 1/a^n

Step 1: identify where the minus sign sits. If the minus is in the index (as in $8^{-5}$ ), proceed to step 2. If the minus is on the base (as in $(-2)^3$ ), that is a different structure: a negative base to an odd power is negative.

Step 2: write one over. Apply $a^{-n} = \frac{1}{a^n}$ . Place the base and its positive index in the denominator. The numerator is 1. The result is a positive fraction.

Step 3: do not swap base and index. The reciprocal of $8^5$ is $\frac{1}{8^5} = 8^{-5}$ . The base stays as 8. The index becomes $-5$ . Writing $5^{-8}$ swaps the digits and is a different expression entirely.

Step 4: sense-check the sign. For a positive base, $a^{-n}$ is always positive and less than 1 when $a > 1$ . If your answer is negative, recheck whether you misread the index minus as a result minus.

Worked example

Evaluate $8^{-5}$ as a fraction.

Rule. $a^{-n} = \frac{1}{a^n}$ , so
$8^{-5} = \frac{1}{8^5} = \frac{1}{32768}$
Trap: $-8^5 = -32768$ , which is large and negative. These differ by a factor of $32768^2$ .

Circle the reciprocal of $8^5$ from the list $8^{-5}$ , $-8^5$ , $5^{-8}$ , $5^8$ .

Reciprocal = one over. $\text{reciprocal of } 8^5 = \frac{1}{8^5} = 8^{-5}$ . Keep the base 8, negate the index.
Reject the traps. $-8^5$ is a sign change, not a reciprocal. $5^{-8}$ and $5^8$ swap the base and index. Only $8^{-5}$ equals $\frac{1}{8^5}$ . ✓

Symbolic: given $7^n = x$ , find $7^{-n}$ .

Apply the rule. $7^{-n} = \frac{1}{7^n}$ .
Substitute. Since $7^n = x$ :
$7^{-n} = \frac{1}{x}$
Trap: $-x$ , which confuses the negative index with a sign change on the value $x$ .

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Common questions

Why does $8^{-5}$ equal a positive number and not minus $8^5$ ?: Because the minus sign is in the index, not in front of the base. The rule $a^{-n} = \frac{1}{a^n}$ says: negative index means one over. So $8^{-5} = \frac{1}{8^5} = \frac{1}{32768}$ , which is positive. The expression $-8^5 = -32768$ is a sign change on the result, a completely different operation.
How do you write the reciprocal of $8^5$ as a single power?: Keep the base 8 and negate the index: the reciprocal of $8^5$ is $\frac{1}{8^5} = 8^{-5}$ . Do not write $5^{-8}$ (swapped base and index) or $-8^5$ (sign change, not reciprocal).
Given $7^n = x$ , why is $7^{-n}$ equal to $\frac{1}{x}$ and not $-x$ ?: Because $7^{-n} = \frac{1}{7^n}$ (negative index means one over), and substituting $7^n = x$ gives $7^{-n} = \frac{1}{x}$ . Writing $-x$ would mean the negative index is a sign change on $x$ , which is not what $a^{-n} = \frac{1}{a^n}$ says.

Related misconceptions

Index laws operate on the index, not the baseApplying an index law by multiplying or changing the base instead of operating on the indices, a companion index-laws misconception on the same Higher topic.
Average speed is not the mean of the speedsAveraging the two speeds for a multi-stage journey when the correct method is total distance divided by total time, a different Higher topic with the same trap structure of a plausible wrong operation.

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