Simplify a^(-5) × a^8 × x^3: Combining Multiple Exponents

Q: Why can't I combine a³ and x³ together?

Because a and x are different bases! The product rule $ a^m \times a^n = a^{m+n} $ only works when the bases are identical. Different bases must remain separate.

Q: What do I do with negative exponents when adding?

Treat negative exponents like negative numbers in addition! So $ a^{-5} \times a^8 $ becomes $ a^{-5+8} = a^3 $. Remember: -5 + 8 = 3.

Q: Can I multiply the coefficients of different variables?

Only if there are numerical coefficients! In $ a^{-5} \times a^8 \times x^3 $, there are no numbers to multiply, so you just apply exponent rules to each base separately.

Q: How do I know when I'm completely done simplifying?

You're done when: (1) All like bases are combined, (2) No negative exponents remain (unless required), and (3) Different bases are clearly separated as factors.

Q: What if I have more than two terms with the same base?

Use the same rule! For example: $ a^2 \times a^{-1} \times a^4 = a^{2+(-1)+4} = a^5 $. Just add all the exponents together for that base.

Exponent Rules with Negative Powers

Reduce the following equation:

$a^{-5}\times a^8\times x^3=$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

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 power laws, the multiplication of powers with equal bases (A)

00:07 equals the same base raised to the sum of the exponents (N+M)

00:10 We'll apply this formula to our exercise, one operation at a time

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

00:20 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:

$a^{-5}\times a^8\times x^3=$

Step-by-step solution

To simplify the given mathematical expression, we'll follow these steps:

Step 1: Apply the Product of Powers Property to terms with the same base. For the expression $a^{-5} \times a^8$ , we use the rule:
$a^m \times a^n = a^{m+n}$ .
Step 2: Add the exponents: $-5 + 8 = 3$ . Therefore, $a^{-5} \times a^8 = a^3$ .
Step 3: Since $x^3$ does not share the base $a$ , it remains as is in the expression.

Therefore, the simplified form of the expression $a^{-5} \times a^8 \times x^3$ is:

$a^3 \times x^3$

Final Answer

$a^3\times x^3$

Key Points to Remember

Essential concepts to master this topic

Product Rule: When multiplying same bases, add the exponents together
Technique: For $a^{-5} \times a^8$ , calculate -5 + 8 = 3
Check: Verify different bases stay separate: $a^3 \times x^3$ cannot combine further ✓

Common Mistakes

Avoid these frequent errors

Adding exponents of different bases
Don't combine $a^3 \times x^3$ into $(ax)^3$ = completely wrong result! Different bases must stay separate in multiplication. Always keep different variable bases as separate factors in your final answer.

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why can't I combine a³ and x³ together?

Because a and x are different bases! The product rule $a^m \times a^n = a^{m+n}$ only works when the bases are identical. Different bases must remain separate.

What do I do with negative exponents when adding?

Treat negative exponents like negative numbers in addition! So $a^{-5} \times a^8$ becomes $a^{-5+8} = a^3$ . Remember: -5 + 8 = 3.

Can I multiply the coefficients of different variables?

Only if there are numerical coefficients! In $a^{-5} \times a^8 \times x^3$ , there are no numbers to multiply, so you just apply exponent rules to each base separately.

How do I know when I'm completely done simplifying?

You're done when: (1) All like bases are combined, (2) No negative exponents remain (unless required), and (3) Different bases are clearly separated as factors.

What if I have more than two terms with the same base?

Use the same rule! For example: $a^2 \times a^{-1} \times a^4 = a^{2+(-1)+4} = a^5$ . Just add all the exponents together for that base.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Exponents Rules questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime