What Is Negative 3 To The Negative 3rd Power? | Solved Clearly

Negative three to the negative third power equals -1/27, since the negative exponent moves the base to the denominator and keeps the sign negative.

That expression looks harder than it is. The tricky part is not the arithmetic. It is the mix of two different “negative” ideas in one line: a negative base and a negative exponent.

Once you separate those two jobs, the answer falls into place fast. In this article, you’ll see the exact value, the rule behind it, the most common mistakes, and a clean way to check your work each time.

What The Expression Means Before You Calculate

The expression is:

(-3)-3

Read it as “negative three raised to the negative third power.” Each part matters:

• -3 is the base.
• -3 (the exponent) tells you how many times the base is used, then flipped into the denominator because the exponent is negative.

A negative exponent does not mean “make the answer negative.” It means “take the reciprocal.” The sign of the answer comes from the base and the exponent being odd or even.

Step-By-Step Calculation

Start with the negative exponent rule:

a-n = 1 / an (when a ≠ 0)

Apply it to (-3)-3:

(-3)-3 = 1 / (-3)3

Now evaluate the positive power:

(-3)3 = (-3) × (-3) × (-3) = 9 × (-3) = -27

So:

1 / (-27) = -1/27

That is the final answer: -1/27.

Why The Answer Is Negative

This part trips people up all the time. The exponent is negative, so many learners expect a negative sign from that alone. That is not what creates the sign.

The sign comes from the base -3 being raised to an odd power. Any negative number raised to an odd exponent stays negative. Any negative number raised to an even exponent turns positive.

Since the exponent magnitude here is 3 (an odd number), (-3)3 is negative. Then taking the reciprocal keeps the sign the same, so the final value stays negative.

Quick Sign Pattern You Can Reuse

Use this pattern with negative bases:

• Odd exponent → negative result
• Even exponent → positive result

That pattern works before and after you apply the reciprocal rule for negative exponents.

Taking Negative 3 To The Negative 3rd Power Without Mixing Rules

A clean habit is to handle the exponent sign first, then the multiplication. That keeps the steps tidy and lowers mistakes.

Method 1: Reciprocal First

1. Rewrite (-3)-3 as 1 / (-3)3.
2. Compute (-3)3 = -27.
3. Write the reciprocal: -1/27.

Method 2: Decimal Form After Fraction Form

Once you have -1/27, you can convert it to a decimal if needed:

-1/27 = -0.037037... (the digits 037 repeat)

Fraction form is usually the preferred final form in algebra unless the problem asks for a decimal.

Common Mistakes That Change The Answer

Most wrong answers come from one of three places: dropping parentheses, misreading the negative exponent, or treating the exponent as subtraction.

Parentheses Matter More Than People Expect

Compare these two expressions:

• (-3)-3 → the base is negative three
• -3-3 → this can be read as the negative of 3-3 in many contexts

In this specific case, both forms end up at -1/27. Still, that will not always happen with other exponents, so it is safer to keep parentheses when the base is negative.

Negative Exponent Is Not “Make It Negative”

3-3 is not -27. It is 1/27. The negative exponent tells you to flip the positive power into the denominator.

Then the base sign decides whether the result is positive or negative.

Exponent Means Repeated Multiplication

(-3)-3 does not mean -3 × -3 and then one more negative sign. It means “take the reciprocal of (-3) multiplied by itself three times.”

If you keep that wording in your head, the structure becomes much easier to handle.

Worked Comparison Table For Similar Exponents

Seeing a few nearby values helps the rule stick. This table shows how sign and reciprocal behavior work together for powers of -3.

Expression	Exact Value	What Changed
(-3)4	81	Even exponent gives a positive result.
(-3)3	-27	Odd exponent keeps the result negative.
(-3)2	9	Two negatives multiply to a positive.
(-3)1	-3	Power of 1 leaves the base unchanged.
(-3)0	1	Any nonzero base to power 0 equals 1.
(-3)-1	-1/3	Negative exponent flips to reciprocal.
(-3)-2	1/9	Reciprocal plus even exponent gives positive.
(-3)-3	-1/27	Reciprocal plus odd exponent gives negative.
(-3)-4	1/81	Reciprocal of a positive power result.

Why Teachers Stress The Exponent Rule

This rule is not just a classroom trick. It keeps exponent laws consistent. If negative exponents did not mean reciprocals, basic exponent patterns would break.

Pattern Check With Division

Start with powers of 3 and move down by dividing by 3 each time:

• 33 = 27
• 32 = 9
• 31 = 3
• 30 = 1
• 3-1 = 1/3
• 3-2 = 1/9
• 3-3 = 1/27

The same pattern applies to -3, with the sign alternating because the base is negative. This is the same exponent law taught in standard algebra references such as OpenStax’s exponents section.

One More Way To Verify The Answer

You can multiply your result by (-3)3 and check whether you get 1:

(-1/27) × (-27) = 1

That confirms the reciprocal relationship is correct.

Where Students Lose Points On Tests

Teachers often mark the setup, not just the final value. A student may know the answer but still lose marks from skipped steps when the topic is exponents.

Write The Reciprocal Step Clearly

Going straight from (-3)-3 to -1/27 is fine in your head. On paper, write the middle step too:

(-3)-3 = 1/(-3)3 = 1/(-27) = -1/27

That line shows you know the rule, not just the answer.

Keep Fraction Form Until The End

Decimals can create rounding issues. If the worksheet later asks you to combine this value with other fractions, staying with -1/27 saves time and avoids errors.

If you need a refresher on the rule itself, Khan Academy’s negative exponents lesson gives a clear walkthrough.

Practice Set To Lock In The Pattern

Try these on your own before checking the answers. They use the same idea, with small changes in sign and exponent parity.

Try These Expressions

• (-2)-3
• (-2)-4
• 4-3
• (-5)-1
• (-1)-3

Expression	Exact Value	Sign Reason
(-2)-3	-1/8	Odd power of a negative base stays negative, then reciprocal.
(-2)-4	1/16	Even power gives positive, then reciprocal.
4-3	1/64	Positive base stays positive.
(-5)-1	-1/5	Power 1 keeps base sign, then reciprocal.
(-1)-3	-1	Reciprocal of -1 is still -1.

Final Answer And A Memory Trick

What Is Negative 3 To The Negative 3rd Power? The value is -1/27.

Use this memory trick: a negative exponent flips, and an odd power of a negative base keeps the minus sign. That one line handles this problem and many others that look just like it.

When you see a power with two negatives again, slow down for one beat, spot the base sign, then apply the reciprocal rule. The expression stops looking messy once you treat each symbol as a separate instruction.