10^{1/4} *40^{1/4}*5^{1/2}

To simplify the expression ${10}^{1/4}\cdot {40}^{1/4}\cdot 5^{1/2}$, we can use the properties of exponents and prime factorization. Step 1: Combine terms with the same exponent The first two terms both have the power of $1/4$. According to the rule $a^n\cdot b^n=(a\cdot b{)}^n$: ${10}^{1/4}\cdot {40}^{1/4}=(10\cdot 40{)}^{1/4}={400}^{1/4}$Step 2: Simplify the resulting base Now we rewrite the expression as: ${400}^{1/4}\cdot 5^{1/2}$Notice that $400$ can be written as ${20}^2$. Substituting this in: $({20}^2{)}^{1/4}\cdot 5^{1/2}={20}^{2\cdot 1/4}\cdot 5^{1/2}={20}^{1/2}\cdot 5^{1/2}$Step 3: Combine terms with the exponent $1/2$ Since both terms are now raised to the power of $1/2$ (which is the same as a square root), we can multiply the bases: ${20}^{1/2}\cdot 5^{1/2}=(20\cdot 5{)}^{1/2}$ ${100}^{1/2}$Step 4: Final calculation The expression ${100}^{1/2}$ is equivalent to the square root of 100: $\sqrt{100}=10$ Final Answer: 10 Would you like me to solve another expression involving fractional exponents or roots?