Evaluate (3×7)^(-4): Negative Exponent Practice Problem

Q: Why don't I split up the base (3×7) when dealing with the negative exponent?

The negative exponent applies to the entire base (3×7) as one unit. Splitting it up would change the problem completely! Keep the base exactly as it is and just flip it using the negative exponent rule.

Q: Does a negative exponent make the answer negative?

No! A negative exponent means reciprocal, not negative. $ (3\times7)^{-4} $ becomes a positive fraction $ \frac{1}{(3\times7)^4} $, not a negative number.

Q: Should I calculate (3×7) first before applying the negative exponent?

You can calculate 3×7 = 21 first, but it's not necessary for this problem. The question asks for the equivalent expression, so keeping it as $ \frac{1}{(3\times7)^4} $ is perfect.

Q: What's wrong with the answer choice that has negative exponents in the fraction?

That choice $ \frac{1}{3^{-6}\times7^{-4}} $ creates double negatives! When you have negative exponents in the denominator, they flip back to positive in the numerator, making it completely different from our original expression.

Q: How can I remember the negative exponent rule?

Think of it as "flip and switch": flip the expression into a fraction (if it's not already) and switch the sign of the exponent from negative to positive. $ x^{-n} $ becomes $ \frac{1}{x^n} $!

Negative Exponents with Product Bases

Insert the corresponding expression:

$\left(3\times7\right)^{-4}=$

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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 the exponent laws, when we have a negative exponent

00:07 We can convert to the reciprocal number and obtain a positive exponent

00:10 We will apply this formula to our exercise

00:13 We'll write the reciprocal number (1 divided by the number)

00:21 Proceed to raise to the positive exponent

00:24 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:

$\left(3\times7\right)^{-4}=$

Step-by-step solution

To solve the expression $(3 \times 7)^{-4}$ , we need to apply the rules for negative exponents:

The expression $(3 \times 7)^{-4}$ can be rewritten using the negative exponent rule, which states that $x^{-n} = \frac{1}{x^n}$ . Applying this rule gives:
$(3 \times 7)^{-4} = \frac{1}{(3 \times 7)^4}$

This simplifies the problem, as now it is expressed in terms of a positive exponent.

Checking our choices, the correct expression is match with choice 3: $\frac{1}{(3 \times 7)^4}$

Thus, $(3 \times 7)^{-4} = \frac{1}{(3 \times 7)^4}$ .

Final Answer

$\frac{1}{\left(3\times7\right)^4}$

Key Points to Remember

Essential concepts to master this topic

Rule: For negative exponents, use $x^{-n} = \frac{1}{x^n}$
Technique: Keep the base unchanged: $(3\times7)^{-4} = \frac{1}{(3\times7)^4}$
Check: The base stays as a product, exponent becomes positive in denominator ✓

Common Mistakes

Avoid these frequent errors

Making the exponent positive without creating a fraction
Don't write (3×7)^{-4} = (3×7)^4 or -(3×7)^4! This ignores the negative exponent rule completely and gives a massive wrong answer. Always use the rule x^{-n} = 1/x^n to flip it into a fraction with positive exponent.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why don't I split up the base (3×7) when dealing with the negative exponent?

The negative exponent applies to the entire base (3×7) as one unit. Splitting it up would change the problem completely! Keep the base exactly as it is and just flip it using the negative exponent rule.

Does a negative exponent make the answer negative?

No! A negative exponent means reciprocal, not negative. $(3\times7)^{-4}$ becomes a positive fraction $\frac{1}{(3\times7)^4}$ , not a negative number.

Should I calculate (3×7) first before applying the negative exponent?

You can calculate 3×7 = 21 first, but it's not necessary for this problem. The question asks for the equivalent expression, so keeping it as $\frac{1}{(3\times7)^4}$ is perfect.

What's wrong with the answer choice that has negative exponents in the fraction?

That choice $\frac{1}{3^{-6}\times7^{-4}}$ creates double negatives! When you have negative exponents in the denominator, they flip back to positive in the numerator, making it completely different from our original expression.

How can I remember the negative exponent rule?

Think of it as "flip and switch": flip the expression into a fraction (if it's not already) and switch the sign of the exponent from negative to positive. $x^{-n}$ becomes $\frac{1}{x^n}$ !

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