Understanding the Exponent to the Zero Power: Why 4^0 = 1

Q: Why does any number to the zero power equal 1?

Think of it as a pattern! Look at $ 4^3 = 64 $, $ 4^2 = 16 $, $ 4^1 = 4 $. Each time we decrease the exponent by 1, we divide by 4. So $ 4^0 = 4 ÷ 4 = 1 $!

Q: Does this work for negative numbers too?

Yes! For example, $ (-3)^0 = 1 $ and $ (-10)^0 = 1 $. The zero exponent rule works for any non-zero number, positive or negative.

Q: What about 0 to the zero power?

That's a special case! $ 0^0 $ is considered undefined in most contexts because it creates mathematical contradictions. Only worry about non-zero bases for now.

Q: How is this different from multiplying by zero?

Don't confuse these! $ 4^0 $ means "4 to the zero power" which equals 1. But $ 4 \times 0 = 0 $. The position of the zero matters!

Q: Can I use this rule with fractions?

Absolutely! For example, $ \left(\frac{3}{5}\right)^0 = 1 $ and $ \left(\frac{7}{2}\right)^0 = 1 $. Any non-zero fraction to the zero power equals 1.

Zero Exponent Rule with Non-Zero Base

$4^0=\text{?}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:02 Alright, let's solve this problem together.

00:06 Remember, any number to the power of zero is one.

00:10 This works, as long as the number isn't zero.

00:14 Now, let's apply this rule to our exercise.

00:18 Great job! That's the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$4^0=\text{?}$

Step-by-step solution

To solve this problem, we need to find the value of $4^0$ .

Step 1: According to the properties of exponents, for any non-zero number $a$ , the zero power $a^0$ is always equal to 1.
Step 2: Here, our base is 4, which is a non-zero number.
Step 3: Applying the zero exponent rule, we find:

$4^0 = 1$

Thus, the answer to the question is $1$ , corresponding to choice 3.

Final Answer

$1$

Key Points to Remember

Essential concepts to master this topic

Rule: Any non-zero number raised to the zero power equals 1
Technique: Apply $a^0 = 1$ directly when base ≠ 0
Check: Verify base is non-zero: 4 ≠ 0, so $4^0 = 1$ ✓

Common Mistakes

Avoid these frequent errors

Thinking the answer equals the base number
Don't assume $4^0 = 4$ = wrong answer of 4! This confuses zero exponent with first power. Always remember that ANY non-zero number to the zero power equals 1, regardless of the base.

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 zero power equal 1?

Think of it as a pattern! Look at $4^3 = 64$ , $4^2 = 16$ , $4^1 = 4$ . Each time we decrease the exponent by 1, we divide by 4. So $4^0 = 4 ÷ 4 = 1$ !

Does this work for negative numbers too?

Yes! For example, $(-3)^0 = 1$ and $(-10)^0 = 1$ . The zero exponent rule works for any non-zero number, positive or negative.

What about 0 to the zero power?

That's a special case! $0^0$ is considered undefined in most contexts because it creates mathematical contradictions. Only worry about non-zero bases for now.

How is this different from multiplying by zero?

Don't confuse these! $4^0$ means "4 to the zero power" which equals 1. But $4 \times 0 = 0$ . The position of the zero matters!

Can I use this rule with fractions?

Absolutely! For example, $\left(\frac{3}{5}\right)^0 = 1$ and $\left(\frac{7}{2}\right)^0 = 1$ . Any non-zero fraction to the zero power equals 1.

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