Simplify and find the value of (18)^((1)/(3))×(768)^((1)/(3))

Question

Simplify and find the value of  (18)^((1)/(3))×(768)^((1)/(3))

Bytelearn · Accepted Answer

Understand the Problem: Understand the problem. We need to simplify the expression (18)^(1/3)×(768)^(1/3) by finding the cube root of each number and then multiplying the results.Simplify Each Term: Simplify each term separately. First, we find the cube root of 18, which is (18)^(1/3). Second, we find the cube root of 768, which is (768)^(1/3).Calculate Cube Root of 18: Calculate the cube root of 18. The cube root of 18 is not a whole number, but we can simplify it by looking for factors of 18 that are perfect cubes. Since 18 = 2 × 9 and 9 is a perfect cube (3^2), we can write: (18)^(1/3) = (2 × 9)^(1/3) = 2^(1/3) × 9^(1/3) = 2^(1/3) × 3Calculate Cube Root of 768: Calculate the cube root of 768. We look for factors of 768 that are perfect cubes. 768 can be factored into 2^8 × 3. Since 2^3 is a perfect cube, we can write: (768)^(1/3) = (2^8 × 3)^(1/3) = (2^3)^(8/3) × 3^(1/3) = 2^(8/3) × 3^(1/3) = 2^(2*4/3) × 3^(1/3) = 4 × 2^(4/3) × 3^(1/3)Combine the Results: Combine the results. Now we multiply the simplified cube roots from Step 3 and Step 4: (2^(1/3) × 3) × (4 × 2^(4/3) × 3^(1/3)) = 4 × 2^(1/3 + 4/3) × 3 × 3^(1/3) = 4 × 2^(5/3) × 3 × 3^(1/3)Simplify Further: Simplify the expression further. Since 2^(5/3) is the same as 2^(1+2/3), which is 2 × 2^(2/3), and 3 × 3^(1/3) is the same as 3^(1+1/3), which is 3^(4/3), we can write: 4 × 2 × 2^(2/3) × 3^(4/3) = 8 × 2^(2/3) × 3^(4/3)Recognize Limitations: Recognize that 2^(2/3) and 3^(4/3) cannot be simplified further without a calculator. Since we cannot simplify 2^(2/3) and 3^(4/3) into whole numbers, we leave the expression as is or use a calculator to find the decimal approximation.Calculate Final Value: Calculate the final value using a calculator (if necessary). If we want to find the decimal approximation of the final value, we would use a calculator to compute 8 × 2^(2/3) × 3^(4/3). However, without a calculator, the simplified form of the expression is the final answer.

Simplify and find the value of $\newline$ $(18)^{\frac{1}{3}} \times (768)^{\frac{1}{3}}$

Full solution

More problems from Multiplication with rational exponents

Related topics

Simplify and find the value of\newline(18)13×(768)13(18)^{\frac{1}{3}} \times (768)^{\frac{1}{3}}(18)31​×(768)31​

Simplify and find the value of $\newline$ $(18)^{\frac{1}{3}} \times (768)^{\frac{1}{3}}$