Simplify 11^10 × 11^11: Multiplying Powers with Same Base

Q: Why do I add the exponents instead of multiplying them?

When multiplying powers with the same base, you're essentially combining repeated multiplication. $ 11^{10} \times 11^{11} $ means 11 multiplied by itself (10+11) = 21 times total!

Q: What if the bases were different numbers?

If the bases are different (like $ 3^4 \times 5^2 $), you cannot combine them using exponent rules. You would need to calculate each power separately first.

Q: Is 11^(10+11) the same as 11^21?

Yes! $ 11^{10+11} $ and $ 11^{21} $ are exactly the same because 10 + 11 = 21. Both represent the simplified form of the original expression.

Q: Can I use this rule for division too?

Yes! For division with same bases, you subtract the exponents. For example: $ 11^{15} ÷ 11^5 = 11^{15-5} = 11^{10} $

Q: What does the answer choice 'a'+b' are correct' mean?

This means that both options a' and b' are mathematically equivalent and correct. Since $ 11^{21} $ and $ 11^{10+11} $ represent the same value, both answers are right!

Exponent Rules with Same Base Multiplication

Simplify the following equation:

$11^{10}\times11^{11}=$

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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)

00:07 equals the same base raised to the sum of exponents (N+M)

00:11 We will apply this formula to our exercise

00:15 We will add up the exponents and raise them to this power

00:22 Let's calculate the sum of the exponents

00:26 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following equation:

$11^{10}\times11^{11}=$

Step-by-step solution

To solve the problem of simplifying the equation $11^{10} \times 11^{11}$ , follow these steps:

Step 1: Identify that the bases are the same (11).
Step 2: Apply the multiplication of powers rule, which states that when multiplying like bases, you add the exponents.
Step 3: Add the exponents: $10 + 11$ .
Step 4: Perform the addition: $10 + 11 = 21$ .
Step 5: Write the expression with the new exponent: $11^{10+11}= 11^{21}$ .

Therefore, the simplified expression is $11^{21}$ . This corresponds to options 1 and 2 being correct as they represent the same expression when evaluating the sum, which is also represented by choice 4 as "a'+b' are correct".

Final Answer

a'+b' are correct

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add the exponents
Technique: $11^{10} \times 11^{11} = 11^{10+11} = 11^{21}$
Check: Verify that $11^{10} \times 11^{11}$ equals $11^{21}$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply 10 × 11 = 110 to get $11^{110}$ ! This confuses the multiplication rule with the power rule and gives an astronomically large wrong answer. 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 I add the exponents instead of multiplying them?

When multiplying powers with the same base, you're essentially combining repeated multiplication. $11^{10} \times 11^{11}$ means 11 multiplied by itself (10+11) = 21 times total!

What if the bases were different numbers?

If the bases are different (like $3^4 \times 5^2$ ), you cannot combine them using exponent rules. You would need to calculate each power separately first.

Is 11^(10+11) the same as 11^21?

Yes! $11^{10+11}$ and $11^{21}$ are exactly the same because 10 + 11 = 21. Both represent the simplified form of the original expression.

Can I use this rule for division too?

Yes! For division with same bases, you subtract the exponents. For example: $11^{15} ÷ 11^5 = 11^{15-5} = 11^{10}$

What does the answer choice 'a'+b' are correct' mean?

This means that both options a' and b' are mathematically equivalent and correct. Since $11^{21}$ and $11^{10+11}$ represent the same value, both answers are right!

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