Final answer:
The expression (6^2(square root)x)(4^9(square root)x) simplifies to 9437184 * x by raising the numbers to their respective powers, multiplying the coefficients, and adding the rational exponents of x. The square root is expressed as a rational exponent of 1/2.
Step-by-step explanation:
The question requires simplifying a mathematical expression using rational exponents. The expression given is (6^2(square root)x)(4^9(square root)x). In order to simplify this, we need to apply the rules of exponents to combine the terms. When multiplying exponential terms with the same base, we add the exponents. To express square roots as rational exponents, we use the fact that the square root of a number can be represented as the number to the 1/2 power.
First, we’ll address the square root of x, which can be written as x^(1/2). Hence the given expression can be rewritten as (6^2 * x^(1/2)) * (4^9 * x^(1/2)). To simplify further, we multiply the coefficients (6^2 and 4^9) and the exponential terms with the same base (x^(1/2) and x^(1/2)). This results in 6^2 * 4^9 * x^(1/2 + 1/2). Adding the exponents of x which are both 1/2 yields x^(1/2 + 1/2) = x^1 = x.
The coefficients can be simplified by raising each number to their respective powers, which gives us 36 * 262144. Multiplying these numbers results in 9437184. So the fully simplified expression with a rational exponent is 9437184 * x.