Evaluate (5×8)^(-5): Negative Exponent Expression Solution

Q: Why does a negative exponent make a fraction?

A negative exponent means 'how many times do I divide by this base?' So $ (5 \times 8)^{-5} $ means divide by $ (5 \times 8) $ five times, which equals $ \frac{1}{(5 \times 8)^5} $.

Q: Should I calculate 5×8 = 40 first?

Not necessarily! The question asks for the expression, not the final number. Keep it as $ \frac{1}{(5 \times 8)^5} $ unless specifically asked to simplify further.

Q: What if I accidentally wrote 1/(5×8)^(-5)?

That would be double negative - you'd flip the fraction twice! Remember: one negative exponent = one flip to reciprocal form. Don't add extra negative signs.

Q: Is this the same as 1/(5^(-5) × 8^(-5))?

No! That's a different expression. $ (5 \times 8)^{-5} $ keeps the product together as one base, while $ 5^{-5} \times 8^{-5} $ applies the exponent to each factor separately.

Q: How do I remember the negative exponent rule?

Think 'flip and drop' - flip to make it a fraction (reciprocal) and drop the negative sign from the exponent. So $ a^{-n} $ becomes $ \frac{1}{a^n} $!

Negative Exponent Rules with Product Bases

Choose the expression that corresponds to the following:

$\left(5\times8\right)^{-5}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to exponent laws, when we have a negative exponent

00:06 Therefore we can convert to the reciprocal number and obtain a positive exponent

00:11 We will apply this formula to our exercise

00:16 We write the reciprocal number (1 divided by the number)

00:19 Proceed to raise to the positive exponent

00:22 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the expression that corresponds to the following:

$\left(5\times8\right)^{-5}=$

Step-by-step solution

To solve the expression $\left(5\times8\right)^{-5}$ , we need to apply the rules of exponents related to negative exponents.

First, let's recall the rule for negative exponents: for any non-zero number $a$ : $a^{-n} = \frac{1}{a^n}$ .

Applying this rule to our expression $\left(5\times8\right)^{-5}$ :

The base $5\times8$ is a product of two numbers 5 and 8.
The exponent is -5, which means we have a negative exponent.
According to the property of negative exponents, we invert the base and change the sign of the exponent:

Thus, $\left(5\times8\right)^{-5} = \frac{1}{\left(5\times8\right)^5}$ .

After applying the rule, we arrive at the expression $\frac{1}{(5 \times 8)^5}$ , which matches the given solution.

Final Answer

$\frac{1}{\left(5\times8\right)^5}$

Key Points to Remember

Essential concepts to master this topic

Rule: A negative exponent means take the reciprocal: a^(-n) = 1/a^n
Technique: Keep the base unchanged: (5×8)^(-5) = 1/(5×8)^5
Check: Verify the base stays as a product, not separate: 1/(40)^5 ✓

Common Mistakes

Avoid these frequent errors

Making the exponent positive without taking reciprocal
Don't just change (5×8)^(-5) to (5×8)^5 = wrong answer! This ignores the negative sign completely and gives a huge positive number instead of a tiny fraction. Always flip to reciprocal form: 1/(5×8)^5 when you see negative exponents.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why does a negative exponent make a fraction?

A negative exponent means 'how many times do I divide by this base?' So $(5 \times 8)^{-5}$ means divide by $(5 \times 8)$ five times, which equals $\frac{1}{(5 \times 8)^5}$ .

Should I calculate 5×8 = 40 first?

Not necessarily! The question asks for the expression, not the final number. Keep it as $\frac{1}{(5 \times 8)^5}$ unless specifically asked to simplify further.

What if I accidentally wrote 1/(5×8)^(-5)?

That would be double negative - you'd flip the fraction twice! Remember: one negative exponent = one flip to reciprocal form. Don't add extra negative signs.

Is this the same as 1/(5^(-5) × 8^(-5))?

No! That's a different expression. $(5 \times 8)^{-5}$ keeps the product together as one base, while $5^{-5} \times 8^{-5}$ applies the exponent to each factor separately.

How do I remember the negative exponent rule?

Think 'flip and drop' - flip to make it a fraction (reciprocal) and drop the negative sign from the exponent. So $a^{-n}$ becomes $\frac{1}{a^n}$ !

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