[tex]$\sqrt{25 x_n a^2} b^2 \times \sqrt[3]{8 x^6 y^3}$[/tex]
Answer (2)
Rewrite the expression using properties of square roots and cube roots.
Simplify the constants: 25 = 5 and 3 8 = 2 .
Simplify the variable terms: a 2 = ∣ a ∣ , 3 x 6 = x 2 , and 3 y 3 = y .
Combine all terms to get the final simplified expression: 10∣ a ∣ b 2 x 2 y x n .
Explanation
Understanding the Problem We are given the expression 25 x n a 2 b 2 × 3 8 x 6 y 3 and we want to simplify it. We will assume that x n , a , b , x , and y are real numbers and that 25 x n a 2 ≥ 0 , which implies x n ≥ 0 .
Rewriting the Expression First, we can rewrite the expression as 25 x n a 2 b 2 × 3 8 3 x 6 3 y 3 .
Simplifying Constants Next, we simplify the constants. We know that 25 = 5 and 3 8 = 2 .
Simplifying Roots Now, we simplify the square root and cube root terms. We have a 2 = ∣ a ∣ , 3 x 6 = x 2 , and 3 y 3 = y .
Substituting Back Substituting these simplified terms back into the expression, we get 5 x n ∣ a ∣ b 2 × 2 x 2 y .
Combining Terms Finally, we combine the terms to obtain the simplified expression: 10∣ a ∣ b 2 x 2 y x n .
Final Answer Therefore, the simplified expression is 10∣ a ∣ b 2 x 2 y x n .
Examples
Simplifying algebraic expressions is a fundamental skill in mathematics with applications in various fields. For instance, in physics, you might encounter complex formulas involving square roots and cube roots when calculating energy levels or wave functions. Simplifying these expressions allows for easier manipulation and computation, leading to a better understanding of the physical phenomena being modeled. Similarly, in engineering, simplifying expressions can help optimize designs and reduce computational complexity in simulations.
To simplify the expression 25 x n a 2 b 2 × 3 8 x 6 y 3 , we separate and simplify the constants and variable terms using properties of roots. The final simplified expression is 10∣ a ∣ b 2 x 2 y x n .
;
