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

Video Solution

Solution Steps

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 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.