Solve (3/280)^(-2): Negative Exponent with Complex Fraction

Q: Why does each number in the denominator get its own negative exponent?

Because of the power of a product rule: $ (abc)^n = a^n \times b^n \times c^n $. When you have multiple factors being raised to a power, each factor gets that power individually.

Q: What's the difference between negative and positive exponents here?

The sign of the exponent doesn't change how you distribute it! Whether it's $ (5 \times 8 \times 7)^2 $ or $ (5 \times 8 \times 7)^{-2} $, you still distribute to get $ 5^{-2} \times 8^{-2} \times 7^{-2} $.

Q: Can I simplify this expression further?

Yes! You could use the negative exponent rule $ a^{-n} = \frac{1}{a^n} $ to rewrite it as $ \frac{1}{3^2} \times \frac{(5 \times 8 \times 7)^2}{1} $, but the given form is the correct answer for this question.

Q: Why isn't the numerator 3² instead of 3⁻²?

Because we're applying the original negative exponent to the entire fraction! The numerator 3 gets the same -2 exponent as the denominator. Only flipping the fraction would change $ 3^{-2} $ to $ 3^2 $.

Q: What if I forget to distribute the exponent to all factors?

You'll get an incomplete expression! Always remember: when a product is raised to a power, every single factor in that product gets that same power. Don't leave any factor behind!

Negative Exponents with Product Denominators

Insert the corresponding expression:

$\left(\frac{3}{5\times8\times7}\right)^{-2}=$

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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 that is raised to the power (N)

00:07 equals a fraction where both the numerator and denominator are raised to the power (N)

00:10 We will apply this formula to our exercise

00:15 Raise both the numerator and denominator to the appropriate power, maintaining parentheses

00:20 According to the laws of exponents, a product raised to the power (N)

00:24 is equal to the product broken down into factors where each factor is raised to the power (N)

00:27 We will apply this formula to our exercise

00:30 Break down each multiplication into factors and raise them to the appropriate power (N)

00:37 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:

$\left(\frac{3}{5\times8\times7}\right)^{-2}=$

Step-by-step solution

To solve this problem, let's break down the expression $\left(\frac{3}{5 \times 8 \times 7}\right)^{-2}$ :

Step 1: Recognize that we have a fraction raised to a negative exponent.
Step 2: Apply the power of a fraction rule: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .
Step 3: Specifically, apply the rule with a negative exponent: $\left(\frac{3}{5 \times 8 \times 7}\right)^{-2} = \frac{3^{-2}}{(5 \times 8 \times 7)^{-2}}$ .
Step 4: Use the negative exponent rule for each element in the expression $(5 \times 8 \times 7)^{-2} = 5^{-2} \times 8^{-2} \times 7^{-2}$ .

This gives us the expression: $\frac{3^{-2}}{5^{-2} \times 8^{-2} \times 7^{-2}}$ .

Therefore, the correct expression is $\frac{3^{-2}}{5^{-2} \times 8^{-2} \times 7^{-2}}$ .

Final Answer

$\frac{3^{-2}}{5^{-2}\times8^{-2}\times7^{-2}}$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply negative exponent to both numerator and denominator separately
Technique: $(5 \times 8 \times 7)^{-2} = 5^{-2} \times 8^{-2} \times 7^{-2}$
Check: Each factor in denominator gets the negative exponent individually ✓

Common Mistakes

Avoid these frequent errors

Applying negative exponent only to the entire product
Don't write $\frac{3^{-2}}{(5 \times 8 \times 7)^{-2}}$ and stop there = incomplete simplification! This misses the crucial step of distributing the exponent. Always distribute the negative exponent to each individual factor: $(5 \times 8 \times 7)^{-2} = 5^{-2} \times 8^{-2} \times 7^{-2}$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does each number in the denominator get its own negative exponent?

Because of the power of a product rule: $(abc)^n = a^n \times b^n \times c^n$ . When you have multiple factors being raised to a power, each factor gets that power individually.

What's the difference between negative and positive exponents here?

The sign of the exponent doesn't change how you distribute it! Whether it's $(5 \times 8 \times 7)^2$ or $(5 \times 8 \times 7)^{-2}$ , you still distribute to get $5^{-2} \times 8^{-2} \times 7^{-2}$ .

Can I simplify this expression further?

Yes! You could use the negative exponent rule $a^{-n} = \frac{1}{a^n}$ to rewrite it as $\frac{1}{3^2} \times \frac{(5 \times 8 \times 7)^2}{1}$ , but the given form is the correct answer for this question.

Why isn't the numerator 3² instead of 3⁻²?

Because we're applying the original negative exponent to the entire fraction! The numerator 3 gets the same -2 exponent as the denominator. Only flipping the fraction would change $3^{-2}$ to $3^2$ .

What if I forget to distribute the exponent to all factors?

You'll get an incomplete expression! Always remember: when a product is raised to a power, every single factor in that product gets that same power. Don't leave any factor behind!

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