Simplify (7×9×6)^-6: Converting Negative Power to Fraction

Q: Does $ (7×9×6)^{-6} $ mean the answer will be negative?

No! The negative sign is on the exponent, not the base. Negative exponents create fractions (reciprocals), but the final result is still positive when all bases are positive.

Q: Can I distribute the negative exponent to each factor inside the parentheses?

Yes, but that's not needed here! You could write $ (7×9×6)^{-6} = 7^{-6} × 9^{-6} × 6^{-6} $, but since we're finding $ \frac{1}{(7×9×6)^{-6}} $, it's simpler to work with the whole expression.

Q: How do I remember which direction the negative exponent moves?

Think "negative exponent = flip position"! If it's in the denominator, it flips to numerator (and becomes positive). If it's in numerator, it flips to denominator.

Q: What's the difference between $ a^{-n} $ and $ -a^n $ ?

$ a^{-n} = \frac{1}{a^n} $ (creates a fraction), while $ -a^n $ means the opposite of $ a^n $. Very different meanings!

Negative Exponents with Reciprocal Rules

Insert the corresponding expression:

$\frac{1}{\left(7\times9\times6\right)^{-6}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Let's simplify this problem together.

00:13 First, we'll use the exponent rules to handle the negative exponents.

00:18 Remember, a negative exponent means we'll find the reciprocal and raise it to the positive power.

00:23 Let's apply this idea to our problem.

00:27 Turn the number into its reciprocal by doing one divided by the number.

00:32 Then, raise this reciprocal to the positive version of the original exponent.

00:37 And that's how we solve this problem. Well done!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{1}{\left(7\times9\times6\right)^{-6}}=$

Step-by-step solution

To solve the expression $\frac{1}{\left(7\times9\times6\right)^{-6}}$ , we will employ exponent rules:

Step 1: Recognize that the expression has a negative exponent in the denominator. By the rule of negative exponents, we have $\left(a^n\right)^{-m} = \frac{1}{\left(a^n\right)^m} = \left(a^n\right)^m$ .
Step 2: Therefore, the negative exponent $-6$ becomes positive when moved from the denominator: $\left(7 \times 9 \times 6\right)^{-6}$ becomes $\left(7 \times 9 \times 6\right)^{6}$ once moved to the numerator.

Thus, rewriting the expression, we get $\left(7 \times 9 \times 6\right)^6$ .

The correct multiple-choice answer is choice 3: $\left(7 \times 9 \times 6\right)^6$ .

Final Answer

$\left(7\times9\times6\right)^6$

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponent in denominator becomes positive in numerator
Technique: $\frac{1}{a^{-n}} = a^n$ flips the base upward
Check: Verify that $\frac{1}{(7×9×6)^{-6}} = (7×9×6)^6$ ✓

Common Mistakes

Avoid these frequent errors

Treating negative exponent as making the entire expression negative
Don't think $(7×9×6)^{-6}$ becomes $-(7×9×6)^6$ ! The negative sign affects the exponent position, not the sign of the result. Always remember that negative exponents create reciprocals, not negative numbers.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does the negative exponent disappear when I move it from denominator to numerator?

The negative exponent doesn't disappear - it gets "canceled" by the reciprocal action! When you have $\frac{1}{a^{-n}}$ , you're taking the reciprocal of a reciprocal, which brings you back to the original base with positive exponent.

Does $(7×9×6)^{-6}$ mean the answer will be negative?

No! The negative sign is on the exponent, not the base. Negative exponents create fractions (reciprocals), but the final result is still positive when all bases are positive.

Can I distribute the negative exponent to each factor inside the parentheses?

Yes, but that's not needed here! You could write $(7×9×6)^{-6} = 7^{-6} × 9^{-6} × 6^{-6}$ , but since we're finding $\frac{1}{(7×9×6)^{-6}}$ , it's simpler to work with the whole expression.

How do I remember which direction the negative exponent moves?

Think "negative exponent = flip position"! If it's in the denominator, it flips to numerator (and becomes positive). If it's in numerator, it flips to denominator.

What's the difference between $a^{-n}$ and $-a^n$ ?

$a^{-n} = \frac{1}{a^n}$ (creates a fraction), while $-a^n$ means the opposite of $a^n$ . Very different meanings!

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Simplify (7×9×6)^-6: Converting Negative Power to Fraction

Negative Exponents with Reciprocal Rules

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why does the negative exponent disappear when I move it from denominator to numerator?

Does $(7×9×6)^{-6}$ mean the answer will be negative?

Can I distribute the negative exponent to each factor inside the parentheses?

How do I remember which direction the negative exponent moves?

What's the difference between $a^{-n}$ and $-a^n$ ?

🌟 Unlock Your Math Potential

More Questions

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Simplify the Expression: 1/(2×3)^(-5) Using Negative Exponents

Simplify the Fraction: 1/(2⁷ × 9⁷ × 5⁷) Expression

Simplify 1/(4^-6 × 7^-6): Negative Exponents in Denominators

Simplify 1/(8⁻⁷ × 9⁻⁷ × 5⁻⁷): Negative Exponent Practice

Simplify (7×9×6)^-6: Converting Negative Power to Fraction

Negative Exponents with Reciprocal Rules

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why does the negative exponent disappear when I move it from denominator to numerator?

Does (7×9×6)−6 (7×9×6)^{-6} (7×9×6)−6 mean the answer will be negative?

Can I distribute the negative exponent to each factor inside the parentheses?

How do I remember which direction the negative exponent moves?

What's the difference between a−n a^{-n} a−n and −an -a^n −an?

🌟 Unlock Your Math Potential

More Questions

Power of a Product

Simplify the Expression: 1/(2×3)^(-5) Using Negative Exponents

Simplify the Fraction: 1/(2⁷ × 9⁷ × 5⁷) Expression

Simplify 1/(4^-6 × 7^-6): Negative Exponents in Denominators

Simplify 1/(8⁻⁷ × 9⁻⁷ × 5⁻⁷): Negative Exponent Practice

Does $(7×9×6)^{-6}$ mean the answer will be negative?

What's the difference between $a^{-n}$ and $-a^n$ ?