Simplify 1/(4×5×6)^6: Complex Fraction with Product Exponent

Q: What does a negative exponent actually mean?

A negative exponent means "take the reciprocal". So $ a^{-n} = \frac{1}{a^n} $. It's not making the number negative - it's flipping it to the other side of a fraction!

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

The negative sign in the exponent is an instruction (flip the fraction), not a sign that affects the final answer. $ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} $, which is positive!

Q: Do I need to calculate 4×5×6 first?

No! The question asks for the equivalent expression, not the numerical value. Keep it as $ (4×5×6)^{-6} $ to match the original format.

Q: How can I remember the negative exponent rule?

Think: "Negative exponent = move to opposite side of fraction line." If it's in the denominator with positive exponent, it moves to numerator with negative exponent!

Q: What if the base has multiple terms like (a+b)?

The rule works the same way! $ \frac{1}{(a+b)^n} = (a+b)^{-n} $. The entire expression in parentheses becomes the base with the negative exponent.

Negative Exponents with Product Bases

Insert the corresponding expression:

$\frac{1}{\left(4\times5\times6\right)^6}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:14 A negative x negative always equals a positive

00:20 Apply the exponent laws in order to simplify the negative exponents

00:24 Convert to the reciprocal number and raise to the power (-1)

00:29 Apply this formula to our exercise

00:33 Convert to the reciprocal number (1 divided by the number)

00:37 Raise to power (-1)

00:41 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:

$\frac{1}{\left(4\times5\times6\right)^6}=$

Step-by-step solution

To solve this problem, we will express the given fraction using the rules of exponents:

Given expression:

$\frac{1}{\left(4\times5\times6\right)^6}$

Applying the rule for negative exponents, $\frac{1}{a^n} = a^{-n}$ , we can rewrite the expression as follows:

Step 1: Recognize that the expression inside the fraction's denominator is $\left(4\times5\times6\right)^6$ .

Step 2: Apply the negative exponent rule:

\frac{1}{\left(4\times5\times6\right)^6} = \left(4\times5\times6\right)^{-6}

Therefore, the equivalent expression is

$\left(4\times5\times6\right)^{-6}$ .

Final Answer

$\left(4\times5\times6\right)^{-6}$

Key Points to Remember

Essential concepts to master this topic

Negative Exponent Rule: $\frac{1}{a^n} = a^{-n}$ for any nonzero base
Technique: Rewrite $\frac{1}{(4×5×6)^6}$ as $(4×5×6)^{-6}$
Check: Convert back to fraction form to verify equivalence ✓

Common Mistakes

Avoid these frequent errors

Adding negative sign to the front
Don't write $-\left(4×5×6\right)^{-6}$ = negative result! The negative exponent rule doesn't create a negative answer, it just flips the fraction. Always apply the rule: $\frac{1}{base^{exponent}} = base^{-exponent}$ without changing signs.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

What does a negative exponent actually mean?

A negative exponent means "take the reciprocal". So $a^{-n} = \frac{1}{a^n}$ . It's not making the number negative - it's flipping it to the other side of a fraction!

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

The negative sign in the exponent is an instruction (flip the fraction), not a sign that affects the final answer. $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$ , which is positive!

Do I need to calculate 4×5×6 first?

No! The question asks for the equivalent expression, not the numerical value. Keep it as $(4×5×6)^{-6}$ to match the original format.

How can I remember the negative exponent rule?

Think: "Negative exponent = move to opposite side of fraction line." If it's in the denominator with positive exponent, it moves to numerator with negative exponent!

What if the base has multiple terms like (a+b)?

The rule works the same way! $\frac{1}{(a+b)^n} = (a+b)^{-n}$ . The entire expression in parentheses becomes the base with the negative exponent.

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