Solve (10×3)/(7×9) Raised to Negative Fourth Power

Video Solution

Solution Steps

00:00 Simplify the following problem

00:04 According to the laws of exponents, a fraction raised to a negative power (-N)

00:09 is equal to its reciprocal raised to the opposite power (N)

00:12 We will apply this formula to our exercise

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

00:32 According to the laws of exponents, a fraction raised to the power (N)

00:36 is equal to the fraction where both the numerator and denominator are raised to the power (N)

00:41 We will apply this formula to our exercise

00:44 We'll raise the numerator and denominator to the appropriate power, maintaining the parentheses

00:49 According to the laws of exponents, a product raised to the power (N)

00:53 is equal to the product broken down into factors where each factor is raised to power (N)

00:57 We will apply this formula to our exercise

01:01 We'll break down each product into factors and raise them to the appropriate power (N)

01:08 This is the solution

Step-by-Step Solution

To solve the problem, let's follow these steps:

Step 1: Recognize that the given expression is $\left(\frac{10 \times 3}{7 \times 9}\right)^{-4}$ . A negative exponent indicates that we should take the reciprocal of the base.
Step 2: Rewrite this expression using the negative exponent rule: $\left(\frac{10 \times 3}{7 \times 9}\right)^{-4} = \left(\frac{7 \times 9}{10 \times 3}\right)^{4}$ This step inverts the fraction and changes the exponent from $-4$ to $4$ .
Step 3: Apply the exponent to each component of the fraction: $\left(\frac{7 \times 9}{10 \times 3}\right)^{4} = \frac{(7 \times 9)^{4}}{(10 \times 3)^{4}}$ This separates the powers between the numerator and the denominator.
Step 4: Distribute the powers inside each product: $= \frac{7^4 \times 9^4}{10^4 \times 3^4}$ This is achieved by applying $(ab)^n = a^n \times b^n$ to both the numerator and the denominator.

Therefore, the simplified expression is $\frac{7^4 \times 9^4}{10^4 \times 3^4}$ , which corresponds to choice 3 in the provided answer choices.