Calculate (4×7)^(-2): Negative Exponent Expression

Q: Why doesn't the negative exponent make the answer negative?

Great question! A negative exponent doesn't mean the answer is negative. It means take the reciprocal. Think of $ 2^{-3} $ as "flip 2³" which gives $ \frac{1}{2^3} = \frac{1}{8} $, not -8!

Q: Can I calculate 4×7 first, then apply the exponent?

Absolutely! You could do $ (4 \times 7)^{-2} = 28^{-2} = \frac{1}{28^2} = \frac{1}{784} $. Both methods give the same answer - use whichever feels easier!

Q: Which rule should I apply first - negative exponent or power of product?

Either order works! You can apply the negative exponent rule first to get $ \frac{1}{(4 \times 7)^2} $, then use power of product. Or do power of product first like in our solution. Math is flexible!

Q: How do I remember what negative exponents mean?

Think "flip it"! When you see a negative exponent, flip the base to the bottom of a fraction and make the exponent positive. $ x^{-2} $ becomes $ \frac{1}{x^2} $.

Q: Why is the correct answer not simplified to one number?

The question asks for the corresponding expression, not the final calculated value. $ \frac{1}{4^2 \times 7^2} $ shows the mathematical structure clearly, while $ \frac{1}{784} $ is just the numerical result.

Power Rules with Product Bases

Insert the corresponding expression:

$\left(4\times7\right)^{-2}=$

❤️ 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

00:07 Let's simplify this problem together.

00:11 Remember, when dealing with negative exponents.

00:15 We switch to the reciprocal, making the exponent positive.

00:20 Let's use this rule in our example now.

00:29 To deal with an exponent over a product.

00:33 We raise each part to that power.

00:36 Let's apply this to our exercise.

00:42 And there you have it! That's our solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(4\times7\right)^{-2}=$

Step-by-step solution

Step 1: We begin by applying the power of a product rule to the expression $\left(4 \times 7\right)^{-2}$ . According to this rule, $(ab)^n = a^n \times b^n$ . Therefore, we have:

$\left(4 \times 7\right)^{-2} = 4^{-2} \times 7^{-2}$

Step 2: Next, we use the negative exponent rule, which states that $a^{-n} = \frac{1}{a^n}$ . Applying this rule to both parts, we get:

$4^{-2} = \frac{1}{4^2}$ and $7^{-2} = \frac{1}{7^2}$

So, $4^{-2} \times 7^{-2} = \frac{1}{4^2} \times \frac{1}{7^2}$

By multiplying these fractions, we obtain:

$\frac{1}{4^2 \times 7^2}$

Therefore, the solution to the problem is $\frac{1}{4^2 \times 7^2}$ .

Keep in mind - we could have used the rules in the other way around, first the negative exponent rule, and only then the product rule and the result would still be the same!

Final Answer

$\frac{1}{4^2\times7^2}$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply negative exponent rule: $a^{-n} = \frac{1}{a^n}$
Technique: Use power of product rule: $(4 \times 7)^{-2} = 4^{-2} \times 7^{-2}$
Check: Verify final answer equals $\frac{1}{28^2} = \frac{1}{784}$ ✓

Common Mistakes

Avoid these frequent errors

Making the entire expression negative
Don't think $(4 \times 7)^{-2} = -(4 \times 7)^2$ = negative result! The negative exponent means reciprocal, not negative sign. Always remember: negative exponent creates a fraction, not a negative number.

Practice Quiz

Test your knowledge with interactive questions

$$ (4^2)^3+(g^3)^4= $$

FAQ

Everything you need to know about this question

Why doesn't the negative exponent make the answer negative?

Great question! A negative exponent doesn't mean the answer is negative. It means take the reciprocal. Think of $2^{-3}$ as "flip 2³" which gives $\frac{1}{2^3} = \frac{1}{8}$ , not -8!

Can I calculate 4×7 first, then apply the exponent?

Absolutely! You could do $(4 \times 7)^{-2} = 28^{-2} = \frac{1}{28^2} = \frac{1}{784}$ . Both methods give the same answer - use whichever feels easier!

Which rule should I apply first - negative exponent or power of product?

Either order works! You can apply the negative exponent rule first to get $\frac{1}{(4 \times 7)^2}$ , then use power of product. Or do power of product first like in our solution. Math is flexible!

How do I remember what negative exponents mean?

Think "flip it"! When you see a negative exponent, flip the base to the bottom of a fraction and make the exponent positive. $x^{-2}$ becomes $\frac{1}{x^2}$ .

Why is the correct answer not simplified to one number?

The question asks for the corresponding expression, not the final calculated value. $\frac{1}{4^2 \times 7^2}$ shows the mathematical structure clearly, while $\frac{1}{784}$ is just the numerical result.

🌟 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