Compare (-1) Raised to Powers 69 and 79: Which Value is Larger?

Negative Base Exponents with Odd Powers

Which is larger?

(−1)69⬜−(−1)79 (-1)^{69}⬜-(-1)^{79}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Place the appropriate sign
00:03 First let's calculate the sign
00:07 Odd power, therefore the sign remains negative
00:12 1 raised to any power always equals 1
00:15 Now let's calculate the sign of the second power
00:20 Odd power, therefore the sign remains negative
00:26 1 raised to any power always equals 1
00:30 Negative times negative always equals positive
00:35 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Which is larger?

(−1)69⬜−(−1)79 (-1)^{69}⬜-(-1)^{79}

2

Step-by-step solution

First, let's evaluate (−1)69(-1)^{69}. Since 69 is an odd number, (−1)69=−1(-1)^{69} = -1.

Next, evaluate −(−1)79-(-1)^{79}. The power 79 is also odd, so (−1)79=−1(-1)^{79} = -1. Applying the additional negative sign results in −(−1)=1-(-1) = 1.

Now, we compare these two results. The expression (−1)69=−1(-1)^{69} = -1 and the expression −(−1)79=1-(-1)^{79} = 1.

Comparing the two: −1-1 is less than 11.

Therefore, the relationship is (−1)69<−(−1)79(-1)^{69} < -(-1)^{79}.

The correct answer is <\lt.

3

Final Answer

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Key Points to Remember

Essential concepts to master this topic
  • Rule: Negative base to odd power equals negative one
  • Technique: (−1)69=−1 (-1)^{69} = -1 and −(−1)79=−(−1)=1 -(-1)^{79} = -(-1) = 1
  • Check: Count exponent parity: odd powers give -1, then apply outer signs ✓

Common Mistakes

Avoid these frequent errors
  • Ignoring the outer negative sign
    Don't treat −(−1)79 -(-1)^{79} as just (−1)79=−1 (-1)^{79} = -1 ! The outer negative changes the final sign from -1 to +1. Always apply the outer negative sign after evaluating the power.

Practice Quiz

Test your knowledge with interactive questions

\( (-2)^7= \)

FAQ

Everything you need to know about this question

Why does (-1) to any odd power always equal -1?

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Odd powers preserve the negative sign! Think of it this way: (−1)3=(−1)×(−1)×(−1)=−1 (-1)^3 = (-1) \times (-1) \times (-1) = -1 . Each pair of negatives becomes positive, but one negative remains unpaired.

What's the difference between (-1)^79 and -(-1)^79?

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The placement of the negative sign matters! (−1)79=−1 (-1)^{79} = -1 because -1 is the base. But −(−1)79=−(−1)=1 -(-1)^{79} = -(-1) = 1 because the negative is outside the power.

How do I remember which powers give positive or negative results?

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Even powers are always positive, odd powers keep the original sign. For (-1): odd powers give -1, even powers give +1. It's like a pattern that alternates!

Why does the outer negative sign matter so much?

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The outer negative is applied after calculating the power. So −(−1)79 -(-1)^{79} means "take the opposite of (−1)79 (-1)^{79} ". Since (−1)79=−1 (-1)^{79} = -1 , the opposite is +1.

Can I just ignore the parentheses around -1?

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Never ignore the parentheses! (−1)69 (-1)^{69} means the entire quantity -1 is raised to the power, while −169 -1^{69} would mean negative of (1 to the 69th power).

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