Evaluate (6×5)^(-3): Negative Exponent Expression Problem

Q: Why can't I just make the exponent positive and keep it on top?

A negative exponent means "reciprocal with positive exponent." $ (6\times5)^{-3} $ literally means "one divided by (6×5) to the 3rd power", not just the positive version.

Q: Do I multiply 6×5 first or apply the exponent rule first?

You can do either! Method 1: $ (6\times5)^{-3} = 30^{-3} = \frac{1}{30^3} $. Method 2: $ (6\times5)^{-3} = 6^{-3} \times 5^{-3} = \frac{1}{6^3 \times 5^3} $. Both give the same answer!

Q: What's the difference between the first and third answer choices?

The first choice $ \frac{1}{(6\times5)^{-3}} $ has a double negative - it's one divided by a negative exponent, which actually makes the result positive again. The correct answer $ \frac{1}{6^3\times5^3} $ properly converts the negative exponent.

Q: How do I remember the negative exponent rule?

Think of it as "flip and switch" - the base flips to the denominator and the exponent switches from negative to positive. $ a^{-n} $ becomes $ \frac{1}{a^n} $.

Q: Is there a way to check my answer without calculating the actual numbers?

Yes! Both $ (6\times5)^{-3} $ and $ \frac{1}{6^3\times5^3} $ should equal $ \frac{1}{30^3} $. Since $ 6\times5 = 30 $ and $ 6^3\times5^3 = (6\times5)^3 = 30^3 $, the expressions are equivalent!

Negative Exponents with Product Rule

Insert the corresponding expression:

$\left(6\times5\right)^{-3}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's simplify this problem together.

00:12 Remember our exponent laws. A negative exponent becomes posit ive by using the reciprocal.

00:18 So, we flip it to a positive exponent by taking one over the n umber.

00:24 We'll use this trick right now.

00:28 Write the reciprocal, that's one divided by the number.

00:32 Then raise it to the positive exponent. Let's keep going!

00:37 To handle an exponent over a product, we expand it.

00:42 Raise each factor to the power.

00:45 Watch as we apply this method.

00:49 And there you have it, the solution is complete!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(6\times5\right)^{-3}=$

Step-by-step solution

To solve the problem of simplifying $(6 \times 5)^{-3}$ , we will proceed as follows:

Step 1: Apply the power of a product rule, which states $(a \times b)^n = a^n \times b^n$ . In our case, apply this to get $(6 \times 5)^{-3} = 6^{-3} \times 5^{-3}$ .
Step 2: Use the negative exponent rule, which is $a^{-n} = \frac{1}{a^n}$ . Applying this to each term, we find:

$6^{-3} = \frac{1}{6^3}$
$5^{-3} = \frac{1}{5^3}$

Step 3: Multiply the results from Step 2:
$6^{-3} \times 5^{-3} = \left(\frac{1}{6^3}\right) \times \left(\frac{1}{5^3}\right) = \frac{1}{6^3 \times 5^3}$ .

Therefore, the expression $(6 \times 5)^{-3}$ simplifies to $\frac{1}{6^3 \times 5^3}$ .

The correct answer choice is:

$\frac{1}{6^3\times5^3}$

Final Answer

$\frac{1}{6^3\times5^3}$

Key Points to Remember

Essential concepts to master this topic

Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$ converts negative powers to fractions
Product Rule: $(a \times b)^{-n} = a^{-n} \times b^{-n} = \frac{1}{a^n \times b^n}$
Check: Substitute back: $(6\times5)^{-3} = 30^{-3} = \frac{1}{30^3} = \frac{1}{27000}$ ✓

Common Mistakes

Avoid these frequent errors

Making the exponent positive but keeping it in the numerator
Don't write $(6\times5)^{-3} = (6\times5)^3$ or leave it as $\frac{1}{(6\times5)^{-3}}$ = wrong structure! The negative exponent means the entire expression moves to the denominator with a positive exponent. Always apply $a^{-n} = \frac{1}{a^n}$ first, then use the product rule.

Practice Quiz

Test your knowledge with interactive questions

$10\cdot10^2\cdot10^{-4}\cdot10^{10}=$

FAQ

Everything you need to know about this question

Why can't I just make the exponent positive and keep it on top?

A negative exponent means "reciprocal with positive exponent." $(6\times5)^{-3}$ literally means "one divided by (6×5) to the 3rd power", not just the positive version.

Do I multiply 6×5 first or apply the exponent rule first?

You can do either! Method 1: $(6\times5)^{-3} = 30^{-3} = \frac{1}{30^3}$ . Method 2: $(6\times5)^{-3} = 6^{-3} \times 5^{-3} = \frac{1}{6^3 \times 5^3}$ . Both give the same answer!

What's the difference between the first and third answer choices?

The first choice $\frac{1}{(6\times5)^{-3}}$ has a double negative - it's one divided by a negative exponent, which actually makes the result positive again. The correct answer $\frac{1}{6^3\times5^3}$ properly converts the negative exponent.

How do I remember the negative exponent rule?

Think of it as "flip and switch" - the base flips to the denominator and the exponent switches from negative to positive. $a^{-n}$ becomes $\frac{1}{a^n}$ .

Is there a way to check my answer without calculating the actual numbers?

Yes! Both $(6\times5)^{-3}$ and $\frac{1}{6^3\times5^3}$ should equal $\frac{1}{30^3}$ . Since $6\times5 = 30$ and $6^3\times5^3 = (6\times5)^3 = 30^3$ , the expressions are equivalent!

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