Express 1/6^7: Converting a Fraction to Negative Exponent Form

Q: Why does the exponent become negative when I move it from denominator to numerator?

The negative exponent rule shows that $ a^{-n} $ and $ \frac{1}{a^n} $ are equivalent expressions. When you move a base from denominator to numerator, the exponent changes sign to maintain equality.

Q: Is $ 6^{-7} $ the same as $ -6^7 $ ?

No, they're completely different! $ 6^{-7} $ is a tiny positive fraction, while $ -6^7 $ is a huge negative number. The negative sign affects the exponent, not the base.

Q: How do I remember which way to convert between fractions and negative exponents?

Think of it as "flip and switch": flip the fraction (move from denominator to numerator) and switch the exponent sign (positive becomes negative or vice versa).

Q: Can I have negative exponents with any base?

Yes! The rule $ a^{-n} = \frac{1}{a^n} $ works for any non-zero base. Just remember that the base itself stays the same - only the exponent changes sign.

Q: What if I see a more complex fraction like $ \frac{3}{2^4} $ ?

You can rewrite this as $ 3 \cdot \frac{1}{2^4} = 3 \cdot 2^{-4} $. The coefficient (3) stays separate, and you only convert the exponential part using negative exponents.

Negative Exponents with Reciprocal Fractions

Insert the corresponding expression:

$\frac{1}{6^7}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's simplify this problem together.

00:09 When a number is raised to a negative power, like negative N.

00:14 It's equal to one over that number raised to the power of N.

00:18 Watch as we use this rule for our exercise.

00:22 We convert it to its reciprocal and multiply by positive one.

00:27 And that's how we find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{1}{6^7}=$

Step-by-step solution

To solve this problem, we will rewrite the expression $\frac{1}{6^7}$ using the rules of exponents:

Step 1: Identify the given fraction.

We start with $\frac{1}{6^7}$ , where the base in the denominator is 6, and the exponent is 7.

Step 2: Apply the formula for negative exponents.

The formula $a^{-n} = \frac{1}{a^n}$ allows us to rewrite a reciprocal power as a negative exponent. This means the expression $\frac{1}{6^7}$ can be rewritten as $6^{-7}$ .

Step 3: Conclude with the answer.

By transforming $\frac{1}{6^7}$ to its equivalent form using negative exponents, the expression becomes $6^{-7}$ .

Therefore, the correct expression is $\boxed{6^{-7}}$ , which corresponds to choice 2 in the given options.

Final Answer

$6^{-7}$

Key Points to Remember

Essential concepts to master this topic

Rule: $a^{-n} = \frac{1}{a^n}$ converts negative exponents to reciprocals
Technique: Move base from denominator to numerator and flip exponent sign
Check: Verify $6^{-7} = \frac{1}{6^7}$ using the definition ✓

Common Mistakes

Avoid these frequent errors

Adding a negative sign to the base when converting
Don't write $\frac{1}{6^7}$ as $-6^7$ or $-6^{-7}$ = completely wrong values! The negative sign goes only on the exponent, not the base. Always remember that only the exponent becomes negative when converting reciprocals.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

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

The negative exponent rule shows that $a^{-n}$ and $\frac{1}{a^n}$ are equivalent expressions. When you move a base from denominator to numerator, the exponent changes sign to maintain equality.

Is $6^{-7}$ the same as $-6^7$ ?

No, they're completely different! $6^{-7}$ is a tiny positive fraction, while $-6^7$ is a huge negative number. The negative sign affects the exponent, not the base.

How do I remember which way to convert between fractions and negative exponents?

Think of it as "flip and switch": flip the fraction (move from denominator to numerator) and switch the exponent sign (positive becomes negative or vice versa).

Can I have negative exponents with any base?

Yes! The rule $a^{-n} = \frac{1}{a^n}$ works for any non-zero base. Just remember that the base itself stays the same - only the exponent changes sign.

What if I see a more complex fraction like $\frac{3}{2^4}$ ?

You can rewrite this as $3 \cdot \frac{1}{2^4} = 3 \cdot 2^{-4}$ . The coefficient (3) stays separate, and you only convert the exponential part using negative exponents.

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Express 1/6^7: Converting a Fraction to Negative Exponent Form

Negative Exponents with Reciprocal Fractions

❤️ 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 exponent become negative when I move it from denominator to numerator?

Is $6^{-7}$ the same as $-6^7$ ?

How do I remember which way to convert between fractions and negative exponents?

Can I have negative exponents with any base?

What if I see a more complex fraction like $\frac{3}{2^4}$ ?

🌟 Unlock Your Math Potential

More Questions

Negative Exponents

Convert the Expression: Finding the Value of 1/3²

Evaluate the Expression: Simplifying 1/4²

Evaluate (10/17)^(-5): Negative Exponent Fraction Problem

Evaluate (5/6)^(-3): Negative Exponent Expression Solution

Powers of a Fraction

Convert the Expression: Writing 1/5² in Standard Form

Simplify the Expression: Converting 1/(20²) to Its Final Form

Evaluate (3/8)^(-5): Negative Exponents with Fractions

Evaluate (1/3)^(-4): Negative Exponent Simplification

Express 1/6^7: Converting a Fraction to Negative Exponent Form

Negative Exponents with Reciprocal Fractions

❤️ 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 exponent become negative when I move it from denominator to numerator?

Is 6−7 6^{-7} 6−7 the same as −67 -6^7 −67?

How do I remember which way to convert between fractions and negative exponents?

Can I have negative exponents with any base?

What if I see a more complex fraction like 324 \frac{3}{2^4} 243​?

🌟 Unlock Your Math Potential

More Questions

Negative Exponents

Convert the Expression: Finding the Value of 1/3²

Evaluate the Expression: Simplifying 1/4²

Evaluate (10/17)^(-5): Negative Exponent Fraction Problem

Evaluate (5/6)^(-3): Negative Exponent Expression Solution

Powers of a Fraction

Convert the Expression: Writing 1/5² in Standard Form

Simplify the Expression: Converting 1/(20²) to Its Final Form

Evaluate (3/8)^(-5): Negative Exponents with Fractions

Evaluate (1/3)^(-4): Negative Exponent Simplification

Is $6^{-7}$ the same as $-6^7$ ?

What if I see a more complex fraction like $\frac{3}{2^4}$ ?