Multiply Powers of 11: Simplifying 11²×11³×11⁴

Exponent Multiplication with Same Base

Simplify the following equation:

112×113×114= 11^2\times11^3\times11^4=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Simplify the following problem
00:03 According to the laws of exponents, the multiplication of exponents with an equal base (A)
00:08 equals the same base raised to the sum of the exponents (N+M)
00:12 We will apply this formula to our exercise
00:18 We'll maintain the base and add up the exponents
00:29 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Simplify the following equation:

112×113×114= 11^2\times11^3\times11^4=

2

Step-by-step solution

To solve this problem, we will simplify the expression 112×113×114 11^2 \times 11^3 \times 11^4 by using the multiplication rule of exponents.

  • Step 1: Identify that all the bases are the same, which is 11.

  • Step 2: Apply the exponent multiplication rule: am×an=am+n a^m \times a^n = a^{m+n} .

Now, apply this rule:

112×113×114=112+3+4 11^2 \times 11^3 \times 11^4 = 11^{2+3+4}

Calculate the sum of the exponents:

2+3+4=9 2 + 3 + 4 = 9

Thus, the expression simplifies to:

119 11^9

Therefore, the simplified version of the expression is:

112+3+4=119 11^{2+3+4} = 11^9

Upon reviewing the choices provided, the correct choice for the simplified expression is choice 3: 112+3+4 11^{2+3+4} .

3

Final Answer

112+3+4 11^{2+3+4}

Key Points to Remember

Essential concepts to master this topic
  • Rule: When multiplying same bases, add the exponents together
  • Technique: 112×113×114=112+3+4=119 11^2 \times 11^3 \times 11^4 = 11^{2+3+4} = 11^9
  • Check: Count exponents match: 2+3+4=9, so 119 11^9 is correct ✓

Common Mistakes

Avoid these frequent errors
  • Multiplying exponents instead of adding them
    Don't multiply the exponents like 2×3×4 = 24! This gives 1124 11^{24} which is completely wrong. The rule breaks down because you're confusing multiplication of powers with power of powers. Always add exponents when multiplying same bases.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why do we add exponents instead of multiplying them?

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Because exponents show repeated multiplication! 112×113 11^2 \times 11^3 means (11×11) × (11×11×11) = five 11's total = 115 11^5 . Adding exponents counts the total multiplications.

What if the bases were different numbers?

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You cannot combine them! The rule only works with identical bases. For example, 23×54 2^3 \times 5^4 stays as is - no simplification possible.

Do I need to calculate the final numerical answer?

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Usually no! 119 11^9 is the simplified form. Computing 285,311,670,611 isn't typically required unless specifically asked for the numerical value.

How is this different from (112)3 (11^2)^3 ?

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That's a power of a power! For (am)n (a^m)^n , you multiply exponents: (112)3=112×3=116 (11^2)^3 = 11^{2 \times 3} = 11^6 . Different rule entirely!

Can I write the answer as 112+3+4 11^{2+3+4} ?

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Yes! Both 112+3+4 11^{2+3+4} and 119 11^9 are correct. The first shows your work, while the second is fully simplified.

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