Simplify the Expression: 2^a × 2^2 Using Exponent Rules

Q: Why do we add exponents when multiplying?

Think of it this way: $ 2^3 \times 2^2 $ means (2×2×2) × (2×2). That's 5 twos multiplied together, which equals $ 2^5 $. So 3 + 2 = 5!

Q: What if the bases are different?

The product rule only works with the same base. For different bases like $ 2^3 \times 3^2 $, you must calculate each part separately: 8 × 9 = 72.

Q: Does this work with variables in both exponents?

Absolutely! $ 2^x \times 2^y = 2^{x+y} $. Just add whatever is in the exponents, whether it's numbers, variables, or expressions.

Q: How is this different from the power rule?

The product rule is for multiplication: $ a^m \times a^n = a^{m+n} $. The power rule is for raising a power to a power: $ (a^m)^n = a^{mn} $. Don't mix them up!

Q: Can I use this rule with negative exponents?

Yes! $ 2^a \times 2^{-3} = 2^{a-3} $. Just remember that adding a negative number is the same as subtracting: a + (-3) = a - 3.

Exponent Rules with Same Base Multiplication

Reduce the following equation:

$2^a\times2^2=$

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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 equal bases (A)

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

00:10 We will apply this formula to our exercise

00:13 We'll maintain the base and add the exponents together

00:16 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:

$2^a\times2^2=$

Step-by-step solution

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

Step 1: Identify the common base
Step 2: Apply the property of exponents
Step 3: Simplify the expression

Now, let's work through each step:

Step 1: The problem gives us the expression $2^a \times 2^2$ . Here, the common base is 2.
Step 2: We'll apply the property of exponents, which states that for the same base, you add the exponents: $b^m \times b^n = b^{m+n}$ . In this case, it will be $2^{a+2}$ .
Step 3: Rewriting the expression using this rule, we get: $2^a \times 2^2 = 2^{a+2}$ .

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

Final Answer

$2^{a+2}$

Key Points to Remember

Essential concepts to master this topic

Product Rule: When multiplying same bases, add the exponents together
Technique: $2^a \times 2^2 = 2^{a+2}$ by adding a + 2
Check: Verify with numbers: $2^3 \times 2^2 = 8 \times 4 = 32 = 2^5$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply the exponents like $2^a \times 2^2 = 2^{2a}$ = wrong answer! This confuses the product rule with the power rule. Always add exponents when multiplying same bases: $2^a \times 2^2 = 2^{a+2}$ .

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 when multiplying?

Think of it this way: $2^3 \times 2^2$ means (2×2×2) × (2×2). That's 5 twos multiplied together, which equals $2^5$ . So 3 + 2 = 5!

What if the bases are different?

The product rule only works with the same base. For different bases like $2^3 \times 3^2$ , you must calculate each part separately: 8 × 9 = 72.

Does this work with variables in both exponents?

Absolutely! $2^x \times 2^y = 2^{x+y}$ . Just add whatever is in the exponents, whether it's numbers, variables, or expressions.

How is this different from the power rule?

The product rule is for multiplication: $a^m \times a^n = a^{m+n}$ . The power rule is for raising a power to a power: $(a^m)^n = a^{mn}$ . Don't mix them up!

Can I use this rule with negative exponents?

Yes! $2^a \times 2^{-3} = 2^{a-3}$ . Just remember that adding a negative number is the same as subtracting: a + (-3) = a - 3.

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