Simplify the Expression: 9^7 × 9^3 × 9^5 Using Exponent Rules

Q: Why do I add exponents instead of multiply them?

The multiplication rule for exponents states that when bases are the same, you add the exponents. Think of it this way: $ 9^2 \times 9^3 = (9 \times 9) \times (9 \times 9 \times 9) = 9^5 $, which is 2 + 3!

Q: What if the bases were different numbers?

If bases are different (like $ 3^2 \times 5^4 $), you cannot combine them using exponent rules. The multiplication rule only works when all terms have the exact same base.

Q: Do I need to calculate the final numerical answer?

Not necessarily! $ 9^{15} $ is a perfectly valid simplified form. However, if asked for the numerical value, you'd need a calculator since $ 9^{15} $ is a very large number.

Q: How do I remember when to add versus multiply exponents?

Multiplication of same bases → add exponentsPower of a power (like $ (9^2)^3 $) → multiply exponentsUse the memory trick: 'Same base multiplication = addition!'

Exponent Rules with Same Base Multiplication

Simplify the following equation:

$9^7\times9^3\times9^5=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:11 Let's simplify this problem.

00:15 Remember, when multiplying exponents with the same base, like A,

00:19 we keep the base and add the exponents together, N plus M.

00:24 Let's apply this to our example.

00:27 Keep the base the same and add up the exponents.

00:34 And there you have it, that's the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following equation:

$9^7\times9^3\times9^5=$

Step-by-step solution

To simplify the expression $9^7 \times 9^3 \times 9^5$ , we will use the multiplication rule for exponents which applies to powers with the same base.

Step 1: Identify that all terms have the same base of 9.
Step 2: Add the exponents together since the bases are the same: $7 + 3 + 5$ .
Step 3: Perform the addition: $7 + 3 + 5 = 15$ .

This results in the expression simplifying to $9^{7+3+5} = 9^{15}$ .

Therefore, the expression $9^7 \times 9^3 \times 9^5$ simplifies to $9^{15}$ .

The correct answer is choice (1): $9^{7+3+5}$ .

Final Answer

$9^{7+3+5}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add exponents
Technique: Identify base 9 in all terms: $9^7 \times 9^3 \times 9^5 = 9^{7+3+5}$
Check: Calculate sum: 7 + 3 + 5 = 15, so answer is $9^{15}$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply the exponents like $9^{7\times3\times5} = 9^{105}$ ! This completely changes the problem and gives an enormous wrong answer. Always add exponents when bases are the same.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I add exponents instead of multiply them?

The multiplication rule for exponents states that when bases are the same, you add the exponents. Think of it this way: $9^2 \times 9^3 = (9 \times 9) \times (9 \times 9 \times 9) = 9^5$ , which is 2 + 3!

What if the bases were different numbers?

If bases are different (like $3^2 \times 5^4$ ), you cannot combine them using exponent rules. The multiplication rule only works when all terms have the exact same base.

Do I need to calculate the final numerical answer?

Not necessarily! $9^{15}$ is a perfectly valid simplified form. However, if asked for the numerical value, you'd need a calculator since $9^{15}$ is a very large number.

How do I remember when to add versus multiply exponents?

Multiplication of same bases → add exponents
Power of a power (like $(9^2)^3$ ) → multiply exponents
Use the memory trick: 'Same base multiplication = addition!'

What if there are more than three terms to multiply?

The same rule applies! Just keep adding all the exponents together. For example: $9^2 \times 9^3 \times 9^4 \times 9^1 = 9^{2+3+4+1} = 9^{10}$

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