Simplify the Product: 3^(5x+1/2) × 7^(5x+1/2) × 5^(5x+1/2)

Reduce the following equation:

$3^{5x+\frac{1}{2}}\times7^{5x+\frac{1}{2}}\times5^{5x+\frac{1}{2}}=$

Video Solution

Solution Steps

00:00 Simplify the following problem

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

00:10 Equals a product where each factor is raised to that same power (N)

00:14 We will apply this formula to our exercise

00:18 Note that the exponent (N) contains an addition operation

00:26 This is the solution to the question

Step-by-Step Solution

To solve this problem, we need to simplify the given product of exponentials. Let's follow the steps:

Step 1: Identify that each factor $3^{5x+\frac{1}{2}}$ , $7^{5x+\frac{1}{2}}$ , and $5^{5x+\frac{1}{2}}$ shares the same exponent $5x + \frac{1}{2}$ .
Step 2: Apply the Power of a Product Rule, which allows us to combine the product of exponents with the same power. The rule states: $a^{m} \times b^{m} \times c^{m} = (a \times b \times c)^{m}$ for $a = 3$ , $b = 7$ , and $c = 5$ .
Step 3: Combine the bases under a single exponent: $3^{5x+\frac{1}{2}} \times 7^{5x+\frac{1}{2}} \times 5^{5x+\frac{1}{2}} = (3 \times 7 \times 5)^{5x+\frac{1}{2}}$

Therefore, the reduced form of the given expression is $\left(3 \times 7 \times 5\right)^{5x+\frac{1}{2}}$ .