Simplify the Expression: 9^(2a) × 9^(2x) × 9^a with Multiple Exponents

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

The product rule for exponents states that $ b^m \times b^n = b^{m+n} $. When you multiply powers with the same base, you add the exponents, not multiply them!

Q: What if the bases were different numbers?

If bases are different (like $ 9^{2a} \times 8^{2x} $), you cannot combine them using exponent rules. The expression would stay as separate terms.

Q: How do I combine like terms in the exponent?

Treat variables like regular algebra: $ 2a + a = 3a $, just like $ 2x + x = 3x $. The coefficient rule applies normally in exponents.

Q: Can I simplify this further than 9^(3a+2x)?

No! $ 9^{3a+2x} $ is the simplest form because the variables a and x are different and cannot be combined further.

Q: What if I wrote 9^(2a+2x+a) as my answer?

That shows the right method but isn't fully simplified! You need to combine like terms: $ 2a + a = 3a $, so the final answer is $ 9^{3a+2x} $.

Exponent Rules with Multiple Variables

Reduce the following equation :

$9^{2a}\times9^{2x}\times9^a=$

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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 powers with equal bases (A)

00:08 equals the same base raised to the sum of the exponents (N+M)

00:11 We will apply this formula to our exercise

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

00:28 Let's group the terms together

00:34 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 :

$9^{2a}\times9^{2x}\times9^a=$

Step-by-step solution

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

Step 1: Recognize that all the terms have the same base, which is 9.
Step 2: Apply the exponent addition rule to combine the exponents.
Step 3: Simplify the result by performing algebraic addition on the exponents.

Now, let's work through each step:
Step 1: The problem gives us the expression $9^{2a} \times 9^{2x} \times 9^a$ . All terms share the same base, which is 9.
Step 2: Using the property of exponents $b^m \times b^n = b^{m+n}$ , add the exponents: $(2a) + (2x) + a$ .
Step 3: Combine like terms: $2a + a + 2x = 3a + 2x$ . So, the expression becomes $9^{3a+2x}$ .

Therefore, the simplified form of the given equation is $9^{3a+2x}$ .

Final Answer

$9^{3a+2x}$

Key Points to Remember

Essential concepts to master this topic

Same Base Rule: When bases are identical, add exponents together
Addition Method: Combine $2a + 2x + a$ to get $3a + 2x$
Verification: Check that all exponents were added, not multiplied ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply exponents like 2a × 2x × a = 4a²x! This completely changes the expression and gives wrong answers. Always add exponents when multiplying terms with the same base: 2a + 2x + a = 3a + 2x.

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?

The product rule for exponents states that $b^m \times b^n = b^{m+n}$ . When you multiply powers with the same base, you add the exponents, not multiply them!

What if the bases were different numbers?

If bases are different (like $9^{2a} \times 8^{2x}$ ), you cannot combine them using exponent rules. The expression would stay as separate terms.

How do I combine like terms in the exponent?

Treat variables like regular algebra: $2a + a = 3a$ , just like $2x + x = 3x$ . The coefficient rule applies normally in exponents.

Can I simplify this further than 9^(3a+2x)?

No! $9^{3a+2x}$ is the simplest form because the variables a and x are different and cannot be combined further.

What if I wrote 9^(2a+2x+a) as my answer?

That shows the right method but isn't fully simplified! You need to combine like terms: $2a + a = 3a$ , so the final answer is $9^{3a+2x}$ .

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