Evaluate This: The Zero Power Rule on Decimal 0.1

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

Think of it as the pattern in exponents: $ 0.1^3 = 0.001 $, $ 0.1^2 = 0.01 $, $ 0.1^1 = 0.1 $. Each time we decrease the exponent by 1, we divide by 0.1. So $ 0.1^0 = 0.1 ÷ 0.1 = 1 $!

Q: Does this rule work for decimal numbers too?

Absolutely! The zero exponent rule works for any non-zero number - whether it's a whole number, fraction, or decimal. $ (0.5)^0 = 1 $, $ (2.7)^0 = 1 $, $ (100.25)^0 = 1 $.

Q: What about zero to the power of zero?

Great question! $ 0^0 $ is actually undefined in most contexts. The zero exponent rule only applies to non-zero bases. Since 0.1 ≠ 0, we can safely use the rule.

Q: How is this different from multiplying by zero?

Don't confuse exponents with multiplication! $ (0.1)^0 $ means "0.1 raised to the power of 0" which equals 1. This is completely different from $ 0.1 \times 0 = 0 $.

Q: Will I always get 1 as the answer?

Only when the exponent is exactly zero and the base is non-zero! If the exponent changes to any other number, you'll get different results: $ (0.1)^1 = 0.1 $, $ (0.1)^2 = 0.01 $.

Zero Exponent Rule with Decimal Bases

$(0.1)^0=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:04 Any number (M) raised to the power of 0 is always equal to 1

00:07 This formula is valid as long as the base is not 0

00:11 We will use this formula in our exercise, we can see that the base is not 0

00:17 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$(0.1)^0=$

Step-by-step solution

To solve this problem, let's apply the Zero Exponent Rule:

Step 1: Identify the base, which is $0.1$ , a non-zero number.
Step 2: Recognize that the exponent is $0$ .
Step 3: Apply the zero exponent rule, which states that any non-zero number raised to the power of zero equals $1$ .

According to the Zero Exponent Rule, we have:

$(0.1)^0 = 1$ .

Therefore, the value of $(0.1)^0$ is $\mathbf{1}$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: Any non-zero number raised to the power of zero equals 1
Technique: Apply $a^0 = 1$ where $a \neq 0$ , so $(0.1)^0 = 1$
Check: Verify that 0.1 is non-zero, then apply zero exponent rule ✓

Common Mistakes

Avoid these frequent errors

Thinking the answer equals the base or zero
Don't assume $(0.1)^0 = 0.1$ or $(0.1)^0 = 0$ ! This ignores the zero exponent rule and gives completely wrong results. Always remember that any non-zero number to the power of zero 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 does any number to the power of zero equal 1?

Think of it as the pattern in exponents: $0.1^3 = 0.001$ , $0.1^2 = 0.01$ , $0.1^1 = 0.1$ . Each time we decrease the exponent by 1, we divide by 0.1. So $0.1^0 = 0.1 ÷ 0.1 = 1$ !

Does this rule work for decimal numbers too?

Absolutely! The zero exponent rule works for any non-zero number - whether it's a whole number, fraction, or decimal. $(0.5)^0 = 1$ , $(2.7)^0 = 1$ , $(100.25)^0 = 1$ .

What about zero to the power of zero?

Great question! $0^0$ is actually undefined in most contexts. The zero exponent rule only applies to non-zero bases. Since 0.1 ≠ 0, we can safely use the rule.

How is this different from multiplying by zero?

Don't confuse exponents with multiplication! $(0.1)^0$ means "0.1 raised to the power of 0" which equals 1. This is completely different from $0.1 \times 0 = 0$ .

Will I always get 1 as the answer?

Only when the exponent is exactly zero and the base is non-zero! If the exponent changes to any other number, you'll get different results: $(0.1)^1 = 0.1$ , $(0.1)^2 = 0.01$ .

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