Compare Exponents: Which is Greater, 0^100 or 100^0?

Exponent Rules with Zero Bases

Which is larger?

0100 ——1000 0^{100}\text{ }_{——}100^0

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Determine which is greater?
00:06 0 to the power of any number is always equal to 1
00:12 Any number to the power of 0 is always equal to 1
00:19 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?

0100 ——1000 0^{100}\text{ }_{——}100^0

2

Step-by-step solution

To solve this problem, we'll follow these steps:

  • Step 1: Evaluate 0100 0^{100} .
  • Step 2: Evaluate 1000 100^0 .
  • Step 3: Compare the values obtained in Step 1 and Step 2.

Now, let's work through each step:

Step 1: Evaluate 0100 0^{100} .
Any non-negative integer power of 0 evaluates to 0. Therefore, 0100=0 0^{100} = 0 .

Step 2: Evaluate 1000 100^0 .
By the zero exponent rule for non-zero bases, 1000=1 100^0 = 1 .

Step 3: Compare the values obtained: 0 0 and 1 1 .
Clearly, 0<1 0 < 1 .

Therefore, 0100 0^{100} is less than 1000 100^0 .

The correct choice is: < <

3

Final Answer

< <

Key Points to Remember

Essential concepts to master this topic
  • Zero Exponent Rule: Any non-zero base raised to power 0 equals 1
  • Technique: 1000=1 100^0 = 1 while 0100=0 0^{100} = 0
  • Check: Verify that 0 < 1, so 0100<1000 0^{100} < 100^0 ✓

Common Mistakes

Avoid these frequent errors
  • Confusing zero exponent rule with zero base
    Don't think 0100=1 0^{100} = 1 because it has an exponent! This mixes up the rules and gives 1 = 1 instead of 0 < 1. Always remember: zero exponent rule applies to NON-ZERO bases, while any positive power of zero equals zero.

Practice Quiz

Test your knowledge with interactive questions

Which of the following is equivalent to the expression below?

\( \)\( 10,000^1 \)

FAQ

Everything you need to know about this question

Why does any number to the power of 0 equal 1?

+

This comes from the pattern of exponents! When you divide powers: 103103=10001000=1 \frac{10^3}{10^3} = \frac{1000}{1000} = 1 , but also 103103=103−3=100 \frac{10^3}{10^3} = 10^{3-3} = 10^0 . So 100=1 10^0 = 1 !

What if the base is 0? Does 00 0^0 equal 1?

+

00 0^0 is actually undefined in most contexts! The zero exponent rule only applies to non-zero bases. For this problem, we have 0100 0^{100} , which clearly equals 0.

How do I remember which rule to use?

+

Look at the base first! If base = 0, the answer is 0 (for positive exponents). If base ≠ 0 and exponent = 0, the answer is 1. Base determines the rule!

Why does 0100=0 0^{100} = 0 ?

+

Think of exponents as repeated multiplication: 0100=0×0×0... 0^{100} = 0 \times 0 \times 0... (100 times). Since zero times anything equals zero, the result is always 0!

Are there any exceptions to these rules?

+

These rules work for all positive integer exponents. The only tricky case is 00 0^0 , which mathematicians handle differently depending on the context. For school math, stick to positive exponents!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Powers - special cases questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

More Questions

Click on any question to see the complete solution with step-by-step explanations

Powers

Calculate Square Area: Finding Area When Side Length is (6-3) cm
Click to solve
Calculate Area: Square with Sides 5 Units Longer
Click to solve

Zero Exponent Rule