How to Teach Math with Interactive Notebooks - Idea Galaxy - Free Printable

Let’s carefully work through the math shown in the image.

We are given this expression on the blue card:

3⁻⁵ = 1 / 3⁵

This is an example of a negative exponent rule.

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Step-by-step reasoning:

The rule for negative exponents is:

> a⁻ⁿ = 1 / aⁿ

That means, if you have a base (like 3) raised to a negative power (like -5), you can rewrite it as 1 divided by that same base raised to the positive version of that power.

So:

3⁻⁵ = 1 / 3⁵

Now let’s calculate what 3⁵ actually is:

3⁵ = 3 × 3 × 3 × 3 × 3
= 9 × 3 × 3 × 3
= 27 × 3 × 3
= 81 × 3
= 243

Therefore:

3⁻⁵ = 1 / 243

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But wait — the question isn’t asking us to compute the decimal or fraction value. It’s showing the *rule* being applied correctly.

Looking at the blue card again:
It says “3⁻⁵ = 1 / 3⁵” — and that is correct according to exponent rules.

There’s also a yellow star-shaped foldable with other exponent rules like:

- a⁰ = 1
- a¹ = a
- aᵐ × aⁿ = aᵐ⁺ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- a⁻ᵐ = 1 / aᵐ ← this matches the blue card!

So everything checks out.

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Since the user didn’t ask a specific question but uploaded materials about exponent rules, and the only explicit equation shown is 3⁻⁵ = 1 / 3⁵, which is correct, we’ll confirm that.

If the task was to verify whether this equation is true — yes, it is.

If the task was to simplify 3⁻⁵ — then the simplified fractional form is 1/243.

But since the blue card shows the intermediate step (not the final number), and no instruction says to compute the value, we should stick to what’s presented.

However, looking more closely — perhaps the student is meant to recognize that 3⁻⁵ equals 1 over 3 to the fifth power, and maybe they’re supposed to write the final simplified fraction?

In many school settings, when they show “3⁻⁵ = 1 / 3⁵”, they expect you to go one step further and compute 3⁵ = 243, so the full answer is 1/243.

Let me double-check:

3¹ = 3
3² = 9
3³ = 27
3⁴ = 81
3⁵ = 243 → ✔

So 3⁻⁵ = 1/243

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Final Answer:
1/243