Solve (3×6)/5 Raised to the Negative Fourth Power: Complete Solution

Q: Why does the negative exponent apply to both 3 and 6 in the numerator?

Because of the power of a product rule: $ (ab)^n = a^n \times b^n $. When you have $ (3 \times 6)^{-4} $, the exponent -4 must be applied to each factor separately!

Q: What happens to the 5 in the denominator?

The 5 also gets the negative exponent! Since we have $ \left(\frac{3\times6}{5}\right)^{-4} $, the entire fraction is raised to the -4 power, so both numerator and denominator are affected.

Q: Can I simplify 3×6 to 18 first?

You could, but the question asks for the expression in terms of the original components. Keep it as $ 3^{-4} \times 6^{-4} $ to match the answer format.

Q: How do negative exponents work with fractions?

Negative exponents flip the fraction: $ \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n $. But here we're distributing the exponent to each part instead of flipping first.

Q: Is there a shortcut for this type of problem?

Remember the distribution rule: when a fraction with multiple terms is raised to a power, every single component gets that power applied to it separately.

Negative Exponents with Compound Fractions

Insert the corresponding expression:

$\left(\frac{3\times6}{5}\right)^{-4}=$

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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 laws of exponents, a fraction raised to the power (N)

00:07 equals the numerator and denominator raised to the same power (N)

00:10 We will apply this formula to our exercise

00:18 According to the laws of exponents when a product is raised to the power (N)

00:22 it is equal to each factor in the product separately raised to the same power (N)

00:26 We will apply this formula to our exercise

00:32 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:

$\left(\frac{3\times6}{5}\right)^{-4}=$

Step-by-step solution

To solve this problem, let's break down the expression and apply the rules of exponents:

Step-by-Step Solution:

Step 1: Understand that the expression given is $\left(\frac{3 \times 6}{5}\right)^{-4}$ .
Step 2: Simplify the fraction $\frac{3 \times 6}{5}$ as a single fraction, which is already given.
Step 3: Use the property of negative exponents: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$ . This lets us convert the expression.
Step 4: Apply the negative exponent $-4$ to each component inside the fraction: $(3 \times 6)^{-4}$ becomes $3^{-4} \times 6^{-4}$ and the denominator $5^{-4}$ .

Combining these steps results in the expression:

$\frac{3^{-4} \times 6^{-4}}{5^{-4}}$

This matches choice 3, which is the correct answer.

Therefore, the solution to the problem is $\frac{3^{-4} \times 6^{-4}}{5^{-4}}$ .

Final Answer

$\frac{3^{-4}\times6^{-4}}{5^{-4}}$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply negative exponents to all parts within parentheses
Technique: $(ab)^{-n} = a^{-n} \times b^{-n}$ distributes the exponent
Check: Each component in numerator and denominator gets the negative exponent ✓

Common Mistakes

Avoid these frequent errors

Not applying exponent to all components
Don't just apply -4 to one part like $\frac{3\times6^{-4}}{5}$ = wrong distribution! This ignores the exponent rules for products and quotients. Always apply the negative exponent to every component: numerator becomes $3^{-4} \times 6^{-4}$ and denominator becomes $5^{-4}$ .

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 apply to both 3 and 6 in the numerator?

Because of the power of a product rule: $(ab)^n = a^n \times b^n$ . When you have $(3 \times 6)^{-4}$ , the exponent -4 must be applied to each factor separately!

What happens to the 5 in the denominator?

The 5 also gets the negative exponent! Since we have $\left(\frac{3\times6}{5}\right)^{-4}$ , the entire fraction is raised to the -4 power, so both numerator and denominator are affected.

Can I simplify 3×6 to 18 first?

You could, but the question asks for the expression in terms of the original components. Keep it as $3^{-4} \times 6^{-4}$ to match the answer format.

How do negative exponents work with fractions?

Negative exponents flip the fraction: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ . But here we're distributing the exponent to each part instead of flipping first.

Is there a shortcut for this type of problem?

Remember the distribution rule: when a fraction with multiple terms is raised to a power, every single component gets that power applied to it separately.

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