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

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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 \( 100^0 \)?

FAQ

Everything you need to know about this question

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

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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?

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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?

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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 ?

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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?

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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!

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