Solve: (1/4×6×9)^(-4) Using Negative Exponent Rules

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

The negative exponent affects the position of terms, not their sign! $ a^{-n} $ means $ \frac{1}{a^n} $, but the result is still positive when the base is positive.

Q: Do I distribute the exponent to each factor in the denominator?

Yes! When you have $ (abc)^{-n} $, it becomes $ a^{-n} \times b^{-n} \times c^{-n} $. Each factor gets its own copy of the exponent.

Q: What happens to the 1 in the numerator?

The 1 also gets the exponent: $ 1^{-4} $. Since $ 1^{-4} = \frac{1}{1^4} = \frac{1}{1} = 1 $, it stays as 1, but we show $ 1^{-4} $ to follow the rule correctly.

Q: Can I simplify this expression further?

The question asks for the expression using negative exponents, so $ \frac{1^{-4}}{4^{-4} \times 6^{-4} \times 9^{-4}} $ is the final answer. Converting to positive exponents would give a different form.

Negative Exponent Rules with Fraction Bases

Insert the corresponding expression:

$\left(\frac{1}{4\times6\times9}\right)^{-4}=$

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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 power laws, a fraction that is raised to the power (N)

00:07 equals a fraction where both the numerator and denominator are raised to the power (N)

00:11 We will apply this formula to our exercise

00:15 We will raise both the numerator and denominator to the appropriate power, maintaining the parentheses

00:22 According to the power laws, a product that is raised to the power (N)

00:26 equals the product broken down into factors where each factor is raised to power (N)

00:29 We will apply this formula to our exercise

00:32 We will breakdown each multiplication operation into factors and raise them to the appropriate power (N)

00:37 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:

$\left(\frac{1}{4\times6\times9}\right)^{-4}=$

Step-by-step solution

To solve this expression, we need to apply the rules of exponents to simplify $\left(\frac{1}{4 \times 6 \times 9}\right)^{-4}$ .

First, using the power of a fraction rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ , we express each term separately as follows:

$(4 \times 6 \times 9)^{-4} = 4^{-4} \times 6^{-4} \times 9^{-4}$

Therefore, the original negative exponent transforms to a multiplication of three positive exponents in the denominator represented as:

$\frac{1^{-4}}{4^{-4} \times 6^{-4} \times 9^{-4}}$

Thus, the corresponding expression in terms of powers with negative exponents is:

$\frac{1^{-4}}{4^{-4} \times 6^{-4} \times 9^{-4}}$

The correct answer is Choice 4.

Final Answer

$\frac{1^{-4}}{4^{-4}\times6^{-4}\times9^{-4}}$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply power rule to both numerator and denominator separately
Technique: $(\frac{1}{abc})^{-n} = \frac{1^{-n}}{a^{-n} \times b^{-n} \times c^{-n}}$
Check: Verify each term has negative exponent: $1^{-4}, 4^{-4}, 6^{-4}, 9^{-4}$ ✓

Common Mistakes

Avoid these frequent errors

Converting negative exponents to positive too early
Don't flip $(\frac{1}{4 \times 6 \times 9})^{-4}$ to $(4 \times 6 \times 9)^4$ immediately = skips the required form! This bypasses showing the negative exponent distribution. Always first apply the power rule to get $\frac{1^{-4}}{4^{-4} \times 6^{-4} \times 9^{-4}}$ , then simplify if needed.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

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

The negative exponent affects the position of terms, not their sign! $a^{-n}$ means $\frac{1}{a^n}$ , but the result is still positive when the base is positive.

Do I distribute the exponent to each factor in the denominator?

Yes! When you have $(abc)^{-n}$ , it becomes $a^{-n} \times b^{-n} \times c^{-n}$ . Each factor gets its own copy of the exponent.

What happens to the 1 in the numerator?

The 1 also gets the exponent: $1^{-4}$ . Since $1^{-4} = \frac{1}{1^4} = \frac{1}{1} = 1$ , it stays as 1, but we show $1^{-4}$ to follow the rule correctly.

Can I simplify this expression further?

The question asks for the expression using negative exponents, so $\frac{1^{-4}}{4^{-4} \times 6^{-4} \times 9^{-4}}$ is the final answer. Converting to positive exponents would give a different form.

Why isn't the answer just (4×6×9)⁴?

While that's mathematically equivalent, the question specifically asks for the expression with negative exponents. We need to show the intermediate step before any simplification.

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