Evaluate (2×6)/(5×7) Raised to the Fourth Power: Fraction Expression

Insert the corresponding expression:

$\left(\frac{2\times6}{5\times7}\right)^4=$

Video Solution

Solution Steps

00:00 Simplify the following problem

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

00:09 equals the numerator and denominator, each raised to the same power (N)

00:10 We will apply this formula to our exercise

00:15 Note that the numerator and denominator are products, we must be careful with the parentheses

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

00:30 equals the product of each factors raised to that power (N)

00:35 We will apply this formula to our exercise

01:01 This is the solution

Step-by-Step Solution

To solve this problem, we will simplify the expression $\left(\frac{2 \times 6}{5 \times 7}\right)^4$ .

Step 1: Apply exponent rules to the expression:
Since $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ , we have: $\left(\frac{2 \times 6}{5 \times 7}\right)^4 = \frac{\left(2 \times 6\right)^4}{\left(5 \times 7\right)^4}$
Step 2: Further break down $\left(2 \times 6\right)^4$ and $\left(5 \times 7\right)^4$ using $(ab)^n = a^n \times b^n$ :
$\left(2 \times 6\right)^4 = 2^4 \times 6^4$ $\left(5 \times 7\right)^4 = 5^4 \times 7^4$
Step 3: Substitute back to see if match with any choices:
$\frac{\left(2 \times 6\right)^4}{\left(5 \times 7\right)^4} = \frac{2^4 \times 6^4}{5^4 \times 7^4}$
Step 4: Compare results with choices:
Both $\frac{\left(2 \times 6\right)^4}{\left(5 \times 7\right)^4}$ and $\frac{2^4 \times 6^4}{5^4 \times 7^4}$ are forms of the same answer, indicating they are all correct simplifications.

Hence, according to the choices provided, all answers are correct.