Simplify each expression:
a) $\frac{\sqrt{192 \hbar^8}}{\sqrt{243 \hbar^4}}=$ $\square$
b) $\frac{\sqrt[3]{270 g}}{\sqrt[3]{80 g^{10}}}=$ $\square$
Answer (1)
Simplify the fraction inside the square root or cube root by finding common factors.
Simplify the variables by using exponent rules.
Take the square root or cube root of each term.
The simplified expressions are 9 8 ℏ 2 and 2 g 3 3 .
Explanation
Problem Analysis We are asked to simplify two expressions involving radicals and variables. We will simplify each expression separately, showing all steps.
Simplifying Expression a a) We want to simplify 243 ℏ 4 192 ℏ 8 . First, we can combine the square roots into a single square root: 243 ℏ 4 192 ℏ 8 = 243 ℏ 4 192 ℏ 8 .Now, we simplify the fraction inside the square root. We can simplify the numbers 243 192 by finding the greatest common divisor of 192 and 243. The prime factorization of 192 is 2 6 ⋅ 3 , and the prime factorization of 243 is 3 5 . Thus, the greatest common divisor is 3. Dividing both numerator and denominator by 3, we get 243 192 = 243 ÷ 3 192 ÷ 3 = 81 64 .Next, we simplify the variables: ℏ 4 ℏ 8 = ℏ 8 − 4 = ℏ 4 .So, we have 243 ℏ 4 192 ℏ 8 = 81 64 ℏ 4 .Now, we take the square root of each term: 81 64 ℏ 4 = 81 64 ℏ 4 = 9 8 ℏ 2 .
Simplifying Expression b b) We want to simplify 3 80 g 10 3 270 g . First, we can combine the cube roots into a single cube root: 3 80 g 10 3 270 g = 3 80 g 10 270 g .Now, we simplify the fraction inside the cube root. We can simplify the numbers 80 270 by dividing both numerator and denominator by 10: 80 270 = 8 27 .Next, we simplify the variables: g 10 g = g 1 − 10 = g − 9 = g 9 1 .So, we have 3 80 g 10 270 g = 3 8 27 ⋅ g 9 1 .Now, we take the cube root of each term: 3 8 27 ⋅ g 9 1 = 3 8 3 27 ⋅ 3 g 9 1 = 2 3 ⋅ g 3 1 = 2 g 3 3 .
Final Answer Therefore, the simplified expressions are: a) 9 8 ℏ 2 b) 2 g 3 3
Examples
Simplifying radical expressions is a fundamental skill in algebra and is used in various fields such as physics and engineering. For example, in physics, you might encounter such expressions when dealing with energy calculations or wave functions. In engineering, these simplifications can arise when analyzing stress and strain in materials or when designing electrical circuits. By simplifying these expressions, we can make calculations easier and gain a better understanding of the underlying relationships.
