Solve (10×3)/(3×9) Raised to Power -4: Complex Fraction Expression

Insert the corresponding expression:

$\left(\frac{10\times3}{3\times9}\right)^{-4}=$

Video Solution

Solution Steps

00:00 Simplify the following problem

00:03 According to the laws of exponents, a fraction raised to a power (N)

00:08 equals the numerator and denominator raised to the same power (N)

00:12 We will apply this formula to our exercise

00:17 We'll raise both the numerator and the denominator to the power (N)

00:24 According to the laws of exponents when the entire product is raised to a power (N)

00:28 it is equal to each factor in the product separately raised to the same power (N)

00:33 We will apply this formula to our exercise

00:46 This is the solution

Step-by-Step Solution

To solve this problem, we need to simplify the expression $\left(\frac{10 \times 3}{3 \times 9}\right)^{-4}$ .

Step 1: Simplify the expression inside the parentheses: $\frac{10 \times 3}{3 \times 9} = \frac{30}{27} = \frac{10}{9}$ .
Step 2: We now have $\left(\frac{10}{9}\right)^{-4}$ .
Step 3: Apply the rule for negative exponents: $\left(\frac{10}{9}\right)^{-4} = \left(\frac{9}{10}\right)^4$ .
Step 4: Apply the power of a quotient rule: $\left(\frac{9}{10}\right)^4 = \frac{9^4}{10^4}$ .
Step 5: Expand using exponentiation properties: $\frac{9^4}{10^4} = \frac{(3^2)^4}{10^4} = \frac{3^8}{10^4}$ .
Step 6: Comparing with the given choices, apply rules uniformly: distribute $4$ across numerators and denominators of the specific expression directly.
Step 7: Write using property: $\left(\frac{10 \times 3}{3 \times 9}\right)^{-4} = \frac{10^{-4}\times3^{-4}}{3^{-4}\times9^{-4}}$ .

Therefore, the correct simplified expression is $\frac{10^{-4}\times3^{-4}}{3^{-4}\times9^{-4}}$ .