Expand the Expression: Converting 10^-1 to Decimal Form

Negative Exponents with Power Expansion

Expand the following expression:

10−1= 10^{-1}=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:06 First, let's find which expressions match the original one.
00:10 Remember, when multiplying powers with the same base, A,
00:15 we keep the base and add the exponents, N plus M.
00:19 Let's use this rule in our example.
00:22 Keep the base and add the exponents together.
00:26 This new expression matches the original one.
00:29 Now, let's simplify the other expressions with the same method.
00:33 This expression doesn't match the original.
00:40 This one doesn't match either.
00:45 And this is not equal to the original too.
00:51 And there you have it, that's the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Expand the following expression:

10−1= 10^{-1}=

2

Step-by-step solution

Let's solve the problem step by step:

The expression given is 10−1 10^{-1} . A negative exponent indicates a reciprocal, so:

10−1=110 10^{-1} = \frac{1}{10}

We can express this as a multiplication form of powers of 10:

Using the property of exponents, specifically the multiplication of powers, we can rewrite:

110=10−11×1010 \frac{1}{10} = 10^{-11} \times 10^{10}

To verify:

  • Apply the rule of exponents: 10−11×1010=10−11+10=10−1 10^{-11} \times 10^{10} = 10^{-11 + 10} = 10^{-1}

  • This confirms the expression is correctly transformed back to 10−1 10^{-1} .

Thus, the expanded expression of 10−1 10^{-1} is:

10−11×1010 10^{-11}\times10^{10}

3

Final Answer

10−11×1010 10^{-11}\times10^{10}

Key Points to Remember

Essential concepts to master this topic
  • Rule: Negative exponents mean reciprocal: 10−1=110 10^{-1} = \frac{1}{10}
  • Technique: Split into product: 10−11×1010=10−11+10=10−1 10^{-11} \times 10^{10} = 10^{-11+10} = 10^{-1}
  • Check: Add exponents in product form: -11 + 10 = -1 ✓

Common Mistakes

Avoid these frequent errors
  • Thinking negative exponent means negative number
    Don't think 10−1 10^{-1} = -10! This confuses the negative sign with the exponent. A negative exponent creates a positive fraction. Always remember negative exponents mean 'flip to reciprocal' not 'make negative'.

Practice Quiz

Test your knowledge with interactive questions

\( 112^0=\text{?} \)

FAQ

Everything you need to know about this question

Why does a negative exponent create a fraction?

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A negative exponent tells you to 'flip' the base to the denominator. Think of it as the opposite of multiplication - instead of 101=10 10^1 = 10 , we get 10−1=110 10^{-1} = \frac{1}{10} .

How do I know which powers of 10 to use in the expansion?

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Look for two exponents that add up to your target! Since we need 10−1 10^{-1} , we found -11 + 10 = -1. There are many correct combinations like 10−5×104 10^{-5} \times 10^{4} .

Is 10−1 10^{-1} the same as 0.1?

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Yes! 10−1=110=0.1 10^{-1} = \frac{1}{10} = 0.1 . Negative powers of 10 create decimals: 10−2=0.01 10^{-2} = 0.01 , 10−3=0.001 10^{-3} = 0.001 , etc.

Why is the answer 10−11×1010 10^{-11} \times 10^{10} and not simpler forms?

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The question asks for a specific expansion format using multiplication of powers. While 110 \frac{1}{10} or 0.1 are correct, 10−11×1010 10^{-11} \times 10^{10} shows the mathematical relationship between exponents.

Can I use any two exponents as long as they add to -1?

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Mathematically yes, but in multiple choice, pick the given option! 10−5×104 10^{-5} \times 10^{4} or 10−2×101 10^{-2} \times 10^{1} would also work since they equal 10−1 10^{-1} .

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