Question

Consider the expression below. (√3)⁵ / (√27)⁻² Explain the mathematical properties you would use to simplify this expression into the form 3ᵃ and describe the process to find the value of a. A numerical answer is required along with an explanation of the method.

Answer 1

To simplify the expression (√3)⁵ / (√27)⁻² into the form 3a, we rewrite square roots as fractional exponents, subtract the exponents when dividing, and arrive at the simplified result 311/2, so a = 11/2.

Step-by-step explanation:

To simplify the expression (√3)⁵ / (√27)⁻² into the form 3a, we can use the properties of exponents and roots to rewrite the square roots as fractional exponents. The square root of a number is the same as raising that number to the 1/2 power (Eq. A.8), so we can express √3 as 31/2 and √27 as 271/2 which is equivalent to 33/2.

Next, we apply the property that when dividing powers with the same base, we subtract the exponents (page 4) to get 35/2 / 3-3. Hence, the expression simplifies to 3(5/2) - (-3) = 35/2 + 3 = 311/2. Therefore, the value of a is 11/2.