Solve (4×5)^(-2): Evaluating Negative Exponents Step-by-Step

Q: Why does the negative exponent go to both 4 and 5?

The power of product rule says that when you raise a product to a power, every factor gets that power! Since $ (4 \times 5)^{-2} $ means the whole product is raised to -2, both 4 and 5 must have the -2 exponent.

Q: What's the difference between the negative sign and negative exponent?

A negative exponent like $ 4^{-2} $ means $ \frac{1}{4^2} $. It's not the same as $ -4^2 $, which means negative 16. The negative is part of the exponent, not the base!

Q: Can I simplify the expression further?

Yes! You could calculate: $ 4^{-2} \times 5^{-2} = \frac{1}{4^2} \times \frac{1}{5^2} = \frac{1}{16} \times \frac{1}{25} = \frac{1}{400} $. But the question asks for the equivalent expression, so $ 4^{-2} \times 5^{-2} $ is the correct answer.

Q: What if the numbers inside were different?

The rule stays the same! For any expression like $ (a \times b)^{-n} $, you get $ a^{-n} \times b^{-n} $. The power of product rule works with any numbers or variables.

Power of Product Rule with Negative Exponents

Insert the corresponding expression:

$\left(4\times5\right)^{-2}=$

❤️ 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

00:04 In order to open parentheses with a multiplication operation and an outside exponent

00:09 In order to open parentheses with a multiplication operation and an outside exponent

00:12 We will apply this formula to our exercise

00:21 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(4\times5\right)^{-2}=$

Step-by-step solution

To solve the problem, we must correctly apply the rules for exponents to distribute the given negative exponent across the factors in the expression:

The expression given is $\left(4 \times 5\right)^{-2}$ . First, we apply the power of a product rule:

$\left(4 \times 5\right)^{-2} = 4^{-2} \times 5^{-2}$

This rule allows us to distribute the negative exponent $-2$ to both factors in the product inside the parentheses. Each factor is affected by the exponent.

Now, let’s verify this against the choices provided:

Choice 3: $4^{-2} \times 5^{-2}$ matches our solution with proper application of the power of a product rule.

Therefore, the correct and corresponding expression is $\boxed{4^{-2}\times5^{-2}}$ .

Final Answer

$4^{-2}\times5^{-2}$

Key Points to Remember

Essential concepts to master this topic

Rule: Distribute the negative exponent to each factor in the product
Technique: Apply $(ab)^{-n} = a^{-n} \times b^{-n}$ to get $4^{-2} \times 5^{-2}$
Check: Both factors have the same negative exponent -2 ✓

Common Mistakes

Avoid these frequent errors

Making the exponent negative for only one factor
Don't write $-4^2 \times 5^{-2}$ or $4^{-2} \times 5^2$ = inconsistent exponents! This breaks the power of product rule and gives wrong results. Always distribute the negative exponent to every factor inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does the negative exponent go to both 4 and 5?

The power of product rule says that when you raise a product to a power, every factor gets that power! Since $(4 \times 5)^{-2}$ means the whole product is raised to -2, both 4 and 5 must have the -2 exponent.

What's the difference between the negative sign and negative exponent?

A negative exponent like $4^{-2}$ means $\frac{1}{4^2}$ . It's not the same as $-4^2$ , which means negative 16. The negative is part of the exponent, not the base!

Can I simplify the expression further?

Yes! You could calculate: $4^{-2} \times 5^{-2} = \frac{1}{4^2} \times \frac{1}{5^2} = \frac{1}{16} \times \frac{1}{25} = \frac{1}{400}$ . But the question asks for the equivalent expression, so $4^{-2} \times 5^{-2}$ is the correct answer.

Why isn't the answer negative?

The negative in the exponent doesn't make the result negative! Negative exponents create fractions (reciprocals), not negative numbers. $4^{-2} = \frac{1}{16}$ is positive.

What if the numbers inside were different?

The rule stays the same! For any expression like $(a \times b)^{-n}$ , you get $a^{-n} \times b^{-n}$ . The power of product rule works with any numbers or variables.

🌟 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