Calculate (3×5×7)/(2×4×6) Raised to Power -2: Complex Fraction Challenge

Q: Why do I need to flip the fraction when the exponent is negative?

A negative exponent means "take the reciprocal." So $ \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n} $. This rule helps turn the negative exponent into a positive one!

Q: Are options B and C really the same thing?

Yes! Option B: $ \frac{2^2×4^2×6^2}{3^2×5^2×7^2} $ and Option C: $ \frac{(2×4×6)^2}{(3×5×7)^2} $ are equivalent because (abc)^2 = a^2×b^2×c^2.

Q: What's the difference between (abc)^(-2) and a^(-2)×b^(-2)×c^(-2)?

They're the same thing! The power of a product rule works for negative exponents too: $ (abc)^{-2} = a^{-2} × b^{-2} × c^{-2} $.

Q: How do I know which form to choose as my final answer?

Both forms are mathematically correct! Choose the one that matches the answer choices given. In this problem, recognizing that both B and C are equivalent is the key insight.

Q: What if I calculated the numerical value instead?

While you could calculate $ \frac{3×5×7}{2×4×6} = \frac{105}{48} $ first, the question asks for the expression form, not the decimal result. Always read what the question is asking for!

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 negative power (-N)

00:08 is equals to its reciprocal raised to the opposite power (N)

00:11 We will apply this formula to our exercise

00:15 We'll convert to the reciprocal and raise to the opposite power

00:23 According to the laws of exponents, a fraction raised to the power (N)

00:26 is equal to the fraction where both the numerator and denominator are raised to the power (N)

00:30 We will apply this formula to our exercise

00:35 We'll raise both the numerator and denominator to the appropriate power, maintaining the parentheses

00:43 According to the laws of exponents, a product raised to the power (N)

00:46 is equal to the product broken down into factors where each factor is raised to the power (N)

00:50 We will apply this formula to our exercise

00:54 We'll break down each product into factors and raise to the appropriate power

01:03 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Insert the corresponding expression:

$\left(\frac{3\times5\times7}{2\times4\times6}\right)^{-2}=$

2

Step-by-step solution

To solve this problem, we'll apply the mathematical rules for exponents and fractions:

Step 1: Identify the structure: The given expression is $\left(\frac{3 \times 5 \times 7}{2 \times 4 \times 6}\right)^{-2}$ .
Step 2: Apply the exponent rule by flipping the fraction due to the negative exponent: $\left(\frac{3 \times 5 \times 7}{2 \times 4 \times 6}\right)^{-2} = \left(\frac{2 \times 4 \times 6}{3 \times 5 \times 7}\right)^{2}$
Step 3: Rewrite using the power of a product rule: $\left(\frac{2 \times 4 \times 6}{3 \times 5 \times 7}\right)^{2} = \frac{(2 \times 4 \times 6)^2}{(3 \times 5 \times 7)^2}$
Step 4: Recognize that this matches the form described by choices. Compare with options: - Option 2: $\frac{2^2 \times 4^2 \times 6^2}{3^2 \times 5^2 \times 7^2}$ - Option 3: $\frac{(2 \times 4 \times 6)^2}{(3 \times 5 \times 7)^2}$ is similar since each factor is squared, aligning with separate sea-lined approaches by recombining within parentheses.

Therefore, Option 2 and Option 3 both correctly represent the expression and the answer is: B+C are correct.

3

Final Answer

B+C are correct

Key Points to Remember

Essential concepts to master this topic

Negative Exponent Rule: a^(-n) = 1/a^n, so flip the fraction
Technique: (3×5×7)/(2×4×6) becomes (2×4×6)/(3×5×7) when raised to -2
Check: Both (a×b×c)^2 and a^2×b^2×c^2 give the same result ✓

Common Mistakes

Avoid these frequent errors

Applying negative exponent only to numerator
Don't apply the negative exponent to just 3×5×7 = 1/(3×5×7)^2 while ignoring the denominator! This forgets that the entire fraction gets the exponent. Always flip the whole fraction first, then apply the positive exponent to both parts.

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why do I need to flip the fraction when the exponent is negative?

+

A negative exponent means "take the reciprocal." So $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$ . This rule helps turn the negative exponent into a positive one!

Are options B and C really the same thing?

+

Yes! Option B: $\frac{2^2×4^2×6^2}{3^2×5^2×7^2}$ and Option C: $\frac{(2×4×6)^2}{(3×5×7)^2}$ are equivalent because (abc)^2 = a^2×b^2×c^2.

What's the difference between (abc)^(-2) and a^(-2)×b^(-2)×c^(-2)?

+

They're the same thing! The power of a product rule works for negative exponents too: $(abc)^{-2} = a^{-2} × b^{-2} × c^{-2}$ .

How do I know which form to choose as my final answer?

+

Both forms are mathematically correct! Choose the one that matches the answer choices given. In this problem, recognizing that both B and C are equivalent is the key insight.

What if I calculated the numerical value instead?

+

While you could calculate $\frac{3×5×7}{2×4×6} = \frac{105}{48}$ first, the question asks for the expression form, not the decimal result. Always read what the question is asking for!

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