Why (1/8)^0 Always Equals One: Understanding Exponential Rules

Q: Why doesn't the zero exponent make the answer zero?

Great question! The zero exponent rule is about the exponent being zero, not the result. Think of it this way: $ a^0 = \frac{a^1}{a^1} = 1 $ for any non-zero a.

Q: Does this rule work for any fraction, not just 1/8?

Absolutely! Any non-zero number (whole numbers, fractions, decimals) raised to the power of 0 equals 1. So $ (\frac{3}{7})^0 = 1 $, $ (0.5)^0 = 1 $, and $ (100)^0 = 1 $.

Q: What happens if the base is zero?

Good catch! $ 0^0 $ is a special case that's undefined in most contexts. The zero exponent rule only applies to non-zero bases. Since $ \frac{1}{8} \neq 0 $, we can safely use the rule.

Q: How can I remember this rule?

Try this pattern: $ 2^3 = 8 $, $ 2^2 = 4 $, $ 2^1 = 2 $. Each time the exponent decreases by 1, we divide by the base. So $ 2^0 = 2^1 ÷ 2 = 1 $!

Q: Is there a way to check my answer?

Yes! Use the division pattern: $ (\frac{1}{8})^0 = \frac{(\frac{1}{8})^1}{(\frac{1}{8})^1} = \frac{\frac{1}{8}}{\frac{1}{8}} = 1 $. This confirms our answer is correct.

Zero Exponent Rule with Fractional Bases

$(\frac{1}{8})^0=\text{?}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:04 Let's solve this problem together.

00:07 Remember, any number to the power of zero equals one.

00:11 This works for any number except zero.

00:15 Now, let's apply this rule to our exercise.

00:18 And there you have it, that's the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$(\frac{1}{8})^0=\text{?}$

Step-by-step solution

To solve the problem, $(\frac{1}{8})^0$ , we utilize the Zero Exponent Rule, which states that any non-zero number raised to the power of zero equals $1$ .

Here's a step-by-step explanation:

Step 1: Identify the base and ensure it is non-zero. In this case, the base is $\frac{1}{8}$ , which is indeed non-zero.
Step 2: Apply the Zero Exponent Rule. According to this rule, $\left(\frac{1}{8}\right)^0 = 1$ .
Step 3: Conclude the result: The expression evaluates to $1$ .

Therefore, the correct answer to the problem $(\frac{1}{8})^0$ is $1$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Zero Exponent Rule: Any non-zero number raised to power 0 equals 1
Technique: Check base is non-zero: $\frac{1}{8} \neq 0$ , so apply rule directly
Check: Verify $(\frac{1}{8})^0 = 1$ using pattern: $(\frac{1}{8})^1 ÷ (\frac{1}{8})^1 = 1$ ✓

Common Mistakes

Avoid these frequent errors

Thinking zero exponent makes the result zero
Don't assume $(\frac{1}{8})^0 = 0$ because of the zero exponent = wrong answer! Zero in the exponent doesn't make the result zero. Always remember: any non-zero base to the power of 0 equals 1, regardless of what the base is.

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 doesn't the zero exponent make the answer zero?

Great question! The zero exponent rule is about the exponent being zero, not the result. Think of it this way: $a^0 = \frac{a^1}{a^1} = 1$ for any non-zero a.

Does this rule work for any fraction, not just 1/8?

Absolutely! Any non-zero number (whole numbers, fractions, decimals) raised to the power of 0 equals 1. So $(\frac{3}{7})^0 = 1$ , $(0.5)^0 = 1$ , and $(100)^0 = 1$ .

What happens if the base is zero?

Good catch! $0^0$ is a special case that's undefined in most contexts. The zero exponent rule only applies to non-zero bases. Since $\frac{1}{8} \neq 0$ , we can safely use the rule.

How can I remember this rule?

Try this pattern: $2^3 = 8$ , $2^2 = 4$ , $2^1 = 2$ . Each time the exponent decreases by 1, we divide by the base. So $2^0 = 2^1 ÷ 2 = 1$ !

Is there a way to check my answer?

Yes! Use the division pattern: $(\frac{1}{8})^0 = \frac{(\frac{1}{8})^1}{(\frac{1}{8})^1} = \frac{\frac{1}{8}}{\frac{1}{8}} = 1$ . This confirms our answer is correct.

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