Expand the Expression: 3^(2a+x+a) Step by Step

Q: Why do we multiply instead of add when expanding exponents?

The exponent rule $ b^{m+n} = b^m \times b^n $ tells us that addition in exponents becomes multiplication of the terms. Think of it as $ 3^2 \times 3^1 = 3^{2+1} = 3^3 $ - we add exponents when multiplying same bases!

Q: Can I combine the 2a and a terms in the exponent first?

Yes! You could simplify $ 2a + x + a = 3a + x $ first, giving $ 3^{3a+x} = 3^{3a} \times 3^x $. But the answer shown keeps them separate, which is also correct.

Q: What's the difference between this and distributing multiplication?

Exponent expansion uses multiplication: $ 3^{a+b} = 3^a \times 3^b $. Distribution uses addition: $ 3(a+b) = 3a + 3b $. Don't mix these rules up!

Q: How do I remember which operation to use?

Remember: same base with added exponents = multiply the terms. If you see $ b^{m+n} $, think "multiply $ b^m \times b^n $" not "add $ b^m + b^n $".

Exponent Rules with Addition in Exponents

Expand the following equation:

$3^{2a+x+a}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Identify which expressions are equal to the original expression

00:03 According to the laws of exponents, multiplying exponents with the same base (A)

00:07 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

00:17 We can observe that this expression equals the original expression

00:22 We'll use the same method in order to simplify the remaining expressions

00:24 In this expression the operation is addition and not multiplication, therefore it's not relevant

00:37 This expression is not equal to the original expression

00:44 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Expand the following equation:

$3^{2a+x+a}=$

Step-by-step solution

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

Step 1: Analyze the given expression's exponent.
Step 2: Apply the exponent addition rule to expand the expression.
Step 3: Identify the correct choice from a set of given options.

Now, let's work through each step:
Step 1: The expression given is $3^{2a + x + a}$ . Here the exponent is $2a + x + a$ .
Step 2: We apply the rule $b^{m+n} = b^m \times b^n$ by rewriting the exponent sum as individual terms: $(2a)$ , $x$ , and $a$ .
Thus, we can rewrite the expression using the property of exponents: $3^{2a + x + a} = 3^{2a} \times 3^x \times 3^a$ .
Step 3: Upon expanding, the solution corresponds to option :

$3^{2a}\times3^x\times3^a$

Therefore, the expanded expression is $3^{2a}\times3^x\times3^a$ .

Final Answer

$3^{2a}\times3^x\times3^a$

Key Points to Remember

Essential concepts to master this topic

Rule: When exponents add, split into multiplication of separate terms
Technique: $3^{2a+x+a} = 3^{2a} \times 3^x \times 3^a$ using $b^{m+n} = b^m \times b^n$
Check: Verify each exponent appears separately with same base multiplied together ✓

Common Mistakes

Avoid these frequent errors

Adding bases instead of multiplying terms
Don't write $3^{2a+x+a} = 3^{2a} + 3^x + 3^a$ = wrong addition! This confuses exponent rules with distributive property. Always use $b^{m+n} = b^m \times b^n$ to multiply terms with 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 multiply instead of add when expanding exponents?

The exponent rule $b^{m+n} = b^m \times b^n$ tells us that addition in exponents becomes multiplication of the terms. Think of it as $3^2 \times 3^1 = 3^{2+1} = 3^3$ - we add exponents when multiplying same bases!

Can I combine the 2a and a terms in the exponent first?

Yes! You could simplify $2a + x + a = 3a + x$ first, giving $3^{3a+x} = 3^{3a} \times 3^x$ . But the answer shown keeps them separate, which is also correct.

What's the difference between this and distributing multiplication?

Exponent expansion uses multiplication: $3^{a+b} = 3^a \times 3^b$ . Distribution uses addition: $3(a+b) = 3a + 3b$ . Don't mix these rules up!

How do I remember which operation to use?

Remember: same base with added exponents = multiply the terms. If you see $b^{m+n}$ , think "multiply $b^m \times b^n$ " not "add $b^m + b^n$ ".

Why doesn't the x in the middle change anything?

The variable x is just another term in the exponent sum. It follows the same rule: $3^x$ becomes one of the multiplied factors, just like $3^{2a}$ and $3^a$ .

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