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

Q: Why does $ \frac{1}{4^{-6} \times 7^{-6}} $ become positive?

When you have a negative exponent in the denominator, it flips to become positive! Think of it as: $ \frac{1}{x^{-n}} = x^n $. The negative cancels out when moving from denominator to numerator.

Q: Can I simplify $ 4^{-6} \times 7^{-6} $ differently?

Yes! You can use the product rule for exponents: $ a^n \times b^n = (a \times b)^n $. So $ 4^{-6} \times 7^{-6} = (4 \times 7)^{-6} = 28^{-6} $.

Q: What's the difference between $ (4 \times 7)^6 $ and $ 4^6 \times 7^6 $ ?

They're actually the same thing! Both equal $ 28^6 $. The power of a product rule works both ways: $ (a \times b)^n = a^n \times b^n $.

Q: Why isn't the answer negative?

The original expression has negative exponents, not negative bases! $ 4^{-6} $ means "one over four to the sixth power", which is positive. Only negative bases create negative results.

Q: How do I check if $ (4 \times 7)^6 $ is correct?

Substitute back! $ \frac{1}{4^{-6} \times 7^{-6}} = \frac{1}{(28)^{-6}} = (28)^6 = (4 \times 7)^6 $ ✓. The math checks out perfectly!

Negative Exponents with Product Rule

Insert the corresponding expression:

$\frac{1}{4^{-6}\times7^{-6}}=$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 A product where each factor is raised to the same power (N)

00:09 Can be converted to parentheses of the entire product raised to the power of the factor (N)

00:13 We will apply this formula to our exercise

00:19 Apply the power laws in order to simplify negative exponents

00:22 Convert to the reciprocal number and raise to the power (-1)

00:25 We will apply this formula to our exercise

00:29 Convert to the reciprocal number (1 divided by the number)

00:33 Raise to the power (-1)

00:36 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:

$\frac{1}{4^{-6}\times7^{-6}}=$

Step-by-step solution

To solve this problem, we will simplify the given expression $\frac{1}{4^{-6}\times7^{-6}}$ by using the properties of exponents.

First, note that both $4^{-6}$ and $7^{-6}$ can be combined under the power of a product property:

$4^{-6} \times 7^{-6} = (4 \times 7)^{-6}$

Then, simplify the expression inside the original fraction:

$\frac{1}{(4 \times 7)^{-6}} = (4 \times 7)^6$

We use the principle that the negative sign in the exponent of a denominator can be inverted to a positive sign in the numerator. Hence, the negative exponent in the denominator becomes positive in the numerator.

Therefore, the correctly corresponding expression to $\frac{1}{4^{-6}\times7^{-6}}$ is $(4 \times 7)^6$ .

Final Answer

$\left(4\times7\right)^6$

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponent in denominator becomes positive in numerator
Technique: Combine $4^{-6} \times 7^{-6} = (4 \times 7)^{-6}$
Check: Final answer has positive exponent and no fractions ✓

Common Mistakes

Avoid these frequent errors

Forgetting to combine bases before applying exponent rules
Don't work with $4^{-6}$ and $7^{-6}$ separately = complicated fractions! This makes the problem much harder and leads to calculation errors. Always combine same exponents first: $4^{-6} \times 7^{-6} = (4 \times 7)^{-6}$ .

Practice Quiz

Test your knowledge with interactive questions

$$ (4^2)^3+(g^3)^4= $$

FAQ

Everything you need to know about this question

Why does $\frac{1}{4^{-6} \times 7^{-6}}$ become positive?

When you have a negative exponent in the denominator, it flips to become positive! Think of it as: $\frac{1}{x^{-n}} = x^n$ . The negative cancels out when moving from denominator to numerator.

Can I simplify $4^{-6} \times 7^{-6}$ differently?

Yes! You can use the product rule for exponents: $a^n \times b^n = (a \times b)^n$ . So $4^{-6} \times 7^{-6} = (4 \times 7)^{-6} = 28^{-6}$ .

What's the difference between $(4 \times 7)^6$ and $4^6 \times 7^6$ ?

They're actually the same thing! Both equal $28^6$ . The power of a product rule works both ways: $(a \times b)^n = a^n \times b^n$ .

Why isn't the answer negative?

The original expression has negative exponents, not negative bases! $4^{-6}$ means "one over four to the sixth power", which is positive. Only negative bases create negative results.

How do I check if $(4 \times 7)^6$ is correct?

Substitute back! $\frac{1}{4^{-6} \times 7^{-6}} = \frac{1}{(28)^{-6}} = (28)^6 = (4 \times 7)^6$ ✓. The math checks out perfectly!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Exponents Rules questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

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

Negative Exponents with Product Rule

❤️ 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 $\frac{1}{4^{-6} \times 7^{-6}}$ become positive?

Can I simplify $4^{-6} \times 7^{-6}$ differently?

What's the difference between $(4 \times 7)^6$ and $4^6 \times 7^6$ ?

Why isn't the answer negative?

How do I check if $(4 \times 7)^6$ is correct?

🌟 Unlock Your Math Potential

More Questions

Power of a Product

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

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

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

Simplify 1/((10×7)^5): Fraction with Compound Base Expression

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

Negative Exponents with Product Rule

❤️ 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 14−6×7−6 \frac{1}{4^{-6} \times 7^{-6}} 4−6×7−61​ become positive?

Can I simplify 4−6×7−6 4^{-6} \times 7^{-6} 4−6×7−6 differently?

What's the difference between (4×7)6 (4 \times 7)^6 (4×7)6 and 46×76 4^6 \times 7^6 46×76?

Why isn't the answer negative?

How do I check if (4×7)6 (4 \times 7)^6 (4×7)6 is correct?

🌟 Unlock Your Math Potential

More Questions

Power of a Product

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

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

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

Simplify 1/((10×7)^5): Fraction with Compound Base Expression

Why does $\frac{1}{4^{-6} \times 7^{-6}}$ become positive?

Can I simplify $4^{-6} \times 7^{-6}$ differently?

What's the difference between $(4 \times 7)^6$ and $4^6 \times 7^6$ ?

How do I check if $(4 \times 7)^6$ is correct?