Simplify the Expression: 10^(a+b) × 10^(a+1) × 10^(b+1)

Q: Why do we add exponents instead of multiplying them?

When you multiply powers with the same base, you're essentially combining repeated multiplication. For example, $ 10^2 \times 10^3 = (10 \times 10) \times (10 \times 10 \times 10) = 10^5 $, which is 2 + 3 = 5.

Q: What if I forget to distribute when adding the exponents?

Write out each step clearly! First identify all exponents: (a+b), (a+1), (b+1). Then combine: a+b+a+1+b+1. Finally group like terms: 2a+2b+2.

Q: Can I factor out the 2 from the final answer?

Yes! $ 10^{2a+2b+2} = 10^{2(a+b+1)} = (10^2)^{a+b+1} = 100^{a+b+1} $. Both forms are correct, but 2a+2b+2 is usually the preferred simplified form.

Q: How do I know I added all the exponents correctly?

Count your variables! The original expression has 2 a's and 2 b's total, plus the constants 1 and 1. Your final answer should have 2a + 2b + 2.

Q: What happens if the bases were different numbers?

If the bases are different (like $ 10^a \times 5^b $), you cannot combine them using exponent rules. The rule only works when all bases are identical!

Exponent Laws with Multiple Base Products

Reduce the following equation:

$10^{a+b}\times10^{a+1}\times10^{b+1}=$

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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 exponent laws, the multiplication of exponents with the same bases (A)

00:06 equals the same base raised to the sum of the exponents (N+M)

00:10 We'll apply this formula to our exercise

00:14 We'll maintain the base and add the exponents together

00:31 Let's group the factors together

00:48 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Reduce the following equation:

$10^{a+b}\times10^{a+1}\times10^{b+1}=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the exponents in the given expression.
Step 2: Use the property of exponents for multiplication by like bases.
Step 3: Add the exponents together and simplify.

Now, let's work through each step:

Step 1: The original expression is $10^{a+b} \times 10^{a+1} \times 10^{b+1}$ .

Step 2: Since the base (10) is the same for all terms, we add the exponents:

(a+b) + (a+1) + (b+1)

Step 3: Simplifying further:

a + b + a + 1 + b + 1 = 2a + 2b + 2

Thus, the expression simplifies to:

10^{2a + 2b + 2}

Therefore, the solution to the problem is $10^{2b+2a+2}$ .

Final Answer

$10^{2b+2a+2}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add the exponents
Technique: Add exponents: (a+b) + (a+1) + (b+1) = 2a+2b+2
Check: Count variables in final exponent: 2a+2b+2 has correct terms ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply the exponents like (a+b)×(a+1)×(b+1) = huge complicated expression! This rule only applies to powers raised to powers, not multiplication of same bases. Always add exponents when multiplying powers with the same base.

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?

When you multiply powers with the same base, you're essentially combining repeated multiplication. For example, $10^2 \times 10^3 = (10 \times 10) \times (10 \times 10 \times 10) = 10^5$ , which is 2 + 3 = 5.

What if I forget to distribute when adding the exponents?

Write out each step clearly! First identify all exponents: (a+b), (a+1), (b+1). Then combine: a+b+a+1+b+1. Finally group like terms: 2a+2b+2.

Can I factor out the 2 from the final answer?

Yes! $10^{2a+2b+2} = 10^{2(a+b+1)} = (10^2)^{a+b+1} = 100^{a+b+1}$ . Both forms are correct, but 2a+2b+2 is usually the preferred simplified form.

How do I know I added all the exponents correctly?

Count your variables! The original expression has 2 a's and 2 b's total, plus the constants 1 and 1. Your final answer should have 2a + 2b + 2.

What happens if the bases were different numbers?

If the bases are different (like $10^a \times 5^b$ ), you cannot combine them using exponent rules. The rule only works when all bases are identical!

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