Simplify the Expression: a^(-3) × a^5 × a^(-4)

Q: Why do we add exponents when multiplying powers?

The product rule says $ a^m \times a^n = a^{m+n} $. Think of it as repeated multiplication: $ a^3 \times a^2 = (a \times a \times a) \times (a \times a) = a^5 $.

Q: How do I handle negative exponents when adding?

Treat negative exponents like negative numbers in addition. So -3 + 5 + (-4) becomes -3 + 5 - 4 = -2. Remember: adding a negative is the same as subtracting!

Q: Are all three answer choices really correct?

Yes! $ a^{-2} $, $ \frac{1}{a^2} $, and $ a^{-3+5-4} $ are all equivalent. The first two are simplified forms, and the third shows the process step.

Q: When should I convert negative exponents to fractions?

Both forms are correct! Use $ a^{-2} $ when working with more exponent rules, or $ \frac{1}{a^2} $ when the problem asks for 'no negative exponents' or you're substituting specific values.

Q: What if there are more than three terms to multiply?

The same rule applies! Just keep adding all the exponents together. For example: $ a^2 \times a^{-1} \times a^3 \times a^{-5} = a^{2+(-1)+3+(-5)} = a^{-1} $.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's simplify this problem.

00:12 Remember, when multiplying powers with the same base, like A,

00:17 you add the exponents. So, it's A to the power of N plus M.

00:22 We'll use this rule in our exercise.

00:25 Keep the base the same, and simply add the exponents together.

00:35 Now, if a number is raised to a negative power, like negative N,

00:40 it's the same as one over that number to the power of positive N.

00:45 Let's apply this rule to our question.

00:48 And that's how we find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Reduce the following equation:

$a^{-3}\times a^5\times a^{-4}=$

2

Step-by-step solution

The given expression is $a^{-3} \times a^5 \times a^{-4}$ .

To simplify, use the product of powers property, which states that when multiplying like bases, you add the exponents:

Apply the rule: $a^{-3} \times a^5 \times a^{-4} = a^{-3+5-4}$ .
Calculate the sum of the exponents: $-3 + 5 - 4 = -2$ .

This simplifies the expression to $a^{-2}$ .

Note that $a^{-2}$ can also be expressed as $\frac{1}{a^2}$ using the property of negative exponents $(a^{-n} = \frac{1}{a^n})$ .

Now, let's evaluate the choices:

Choice 1: $a^{-2}$ , which matches our simplification.
Choice 2: $\frac{1}{a^2}$ , which is another form of $a^{-2}$ .
Choice 3: $a^{-3+5-4}$ , which correctly reflects the intermediate step we computed.
Choice 4 states "All answers are correct," which, considering all interpretations and simplifications are valid, is indeed true.

Therefore, the correct answer is: All answers are correct, as each choice corresponds to a logical step or equivalent expression in reducing the equation.

3

Final Answer

All answers are correct

Key Points to Remember

Essential concepts to master this topic

Product Rule: When multiplying same bases, add all exponents together
Technique: $a^{-3} \times a^5 \times a^{-4} = a^{-3+5+(-4)} = a^{-2}$
Check: Verify $a^{-2} = \frac{1}{a^2}$ and intermediate step $a^{-3+5-4}$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply the exponents like (-3) × 5 × (-4) = 60, giving a^60! This completely ignores the product rule. When bases are the same, you multiply by adding exponents, not by multiplying them. Always add: -3 + 5 + (-4) = -2.

Practice Quiz

Test your knowledge with interactive questions

$$

Simplify the following equation:

$5^8\times5^3=$

FAQ

Everything you need to know about this question

Why do we add exponents when multiplying powers?

+

The product rule says $a^m \times a^n = a^{m+n}$ . Think of it as repeated multiplication: $a^3 \times a^2 = (a \times a \times a) \times (a \times a) = a^5$ .

How do I handle negative exponents when adding?

+

Treat negative exponents like negative numbers in addition. So -3 + 5 + (-4) becomes -3 + 5 - 4 = -2. Remember: adding a negative is the same as subtracting!

Are all three answer choices really correct?

+

Yes! $a^{-2}$ , $\frac{1}{a^2}$ , and $a^{-3+5-4}$ are all equivalent. The first two are simplified forms, and the third shows the process step.

When should I convert negative exponents to fractions?

+

Both forms are correct! Use $a^{-2}$ when working with more exponent rules, or $\frac{1}{a^2}$ when the problem asks for 'no negative exponents' or you're substituting specific values.

What if there are more than three terms to multiply?

+

The same rule applies! Just keep adding all the exponents together. For example: $a^2 \times a^{-1} \times a^3 \times a^{-5} = a^{2+(-1)+3+(-5)} = a^{-1}$ .

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Simplify the Expression: a^(-3) × a^5 × a^(-4)

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