Simplify Scientific Notation: 10×10^-3×10^5 Multiplication Problem

Exponent Rules with Scientific Notation

Reduce the following equation:

10×10−3×105= 10\times10^{-3}\times10^5=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:08 Let's simplify this problem together.
00:12 Remember, any number raised to the power of one is just the numbe r itself.
00:18 Let's apply this idea to our exercise by raising it to the power o f one.
00:23 According to the laws of exponents, multiplying powers with the same base, A,
00:29 equals A raised to the sum of the exponents, N plus M.
00:35 We'll use this formula in our exercise step by step.
00:39 Notice that we're adding a negative factor. So, a positive times a negative is always negative.
00:46 Therefore, we subtract as follows.
00:56 And that's how we find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Reduce the following equation:

10×10−3×105= 10\times10^{-3}\times10^5=

2

Step-by-step solution

To simplify the equation 10×10−3×105 10 \times 10^{-3} \times 10^5 , we will apply the exponent multiplication rule which states that when multiplying like bases, we add the exponents.

  • Step 1: Identify exponents on each term - The base for all terms is 1010.
  • Step 2: The expression can be rewritten using implied exponents:
    101×10−3×105 10^1 \times 10^{-3} \times 10^5
  • Step 3: Apply the rule of exponents. When multiplying terms with the same base, add the exponents:
    101+(−3)+5 10^{1 + (-3) + 5}
  • Step 4: Calculate the sum of the exponents:
    1+(−3)+5=3 1 + (-3) + 5 = 3
  • Step 5: Rewrite the expression with the summed exponent:
    103 10^3

Therefore, the simplified expression is 103 10^3 , which is choice 4.

3

Final Answer

103 10^3

Key Points to Remember

Essential concepts to master this topic
  • Rule: When multiplying same bases, add the exponents together
  • Technique: Rewrite 10 10 as 101 10^1 , then add: 1 + (-3) + 5 = 3
  • Check: Verify 103=1000 10^3 = 1000 matches 10×0.001×100000=1000 10 \times 0.001 \times 100000 = 1000 ✓

Common Mistakes

Avoid these frequent errors
  • Multiplying exponents instead of adding them
    Don't multiply the exponents like 1 × (-3) × 5 = -15! This gives 10−15 10^{-15} which is completely wrong. Always add exponents when multiplying powers with the same base.

Practice Quiz

Test your knowledge with interactive questions

Simplify the following equation:

\( \)\( 4^5\times4^5= \)

FAQ

Everything you need to know about this question

Why do I write 10 as 101 10^1 ?

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Any number without a visible exponent has an implied exponent of 1. Writing 101 10^1 makes it easier to see all exponents and apply the addition rule correctly.

What if one of the exponents is negative?

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Still add them! Remember that adding a negative number is the same as subtracting. So 1 + (-3) + 5 becomes 1 - 3 + 5 = 3.

How do I remember when to add vs multiply exponents?

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Multiplying powers with same base = add exponents
Raising a power to a power = multiply exponents
This problem has multiplication, so we add!

Can I check my answer without a calculator?

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Yes! Convert to regular numbers: 101=10 10^1 = 10 , 10−3=0.001 10^{-3} = 0.001 , 105=100000 10^5 = 100000 . Then 10×0.001×100000=1000=103 10 \times 0.001 \times 100000 = 1000 = 10^3 .

What does 103 10^3 mean in real numbers?

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103=10×10×10=1000 10^3 = 10 \times 10 \times 10 = 1000 . The exponent tells you how many times to multiply 10 by itself!

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