Simplify each expression:
a) $\frac{\sqrt{45}}{\sqrt{320}}=$ $\square$
b) $\frac{\sqrt[3]{1250}}{\sqrt[3]{640}}=$ $\square$
Answer (1)
Simplify the first expression by factoring out perfect squares from the square roots: 320 45 = 64 × 5 9 × 5 = 8 5 3 5 .
Cancel the common factor 5 to get 8 3 .
Simplify the second expression by factoring out perfect cubes from the cube roots: 3 640 3 1250 = 3 64 × 10 3 125 × 10 = 4 3 10 5 3 10 .
Cancel the common factor 3 10 to get 4 5 .
The simplified expressions are 8 3 and 4 5 .
Explanation
Understanding the Problem We are given two expressions to simplify: a) 320 45 b) 3 640 3 1250
Simplifying the First Expression To simplify the first expression, we simplify the square roots individually and then simplify the fraction. 320 45 = 64 × 5 9 × 5 = 64 × 5 9 × 5 = 8 5 3 5 Since 5 is a common factor in both the numerator and the denominator, we can cancel it out: 8 5 3 5 = 8 3 Thus, the simplified form of the first expression is 8 3 .
Simplifying the Second Expression To simplify the second expression, we simplify the cube roots individually and then simplify the fraction. 3 640 3 1250 = 3 64 × 10 3 125 × 10 = 3 64 × 3 10 3 125 × 3 10 = 4 3 10 5 3 10 Since 3 10 is a common factor in both the numerator and the denominator, we can cancel it out: 4 3 10 5 3 10 = 4 5 Thus, the simplified form of the second expression is 4 5 .
Final Answer Therefore, the simplified expressions are: a) 320 45 = 8 3 b) 3 640 3 1250 = 4 5
Examples
Simplifying radicals is a fundamental skill in algebra and is useful in various fields. For example, in physics, when calculating the energy of a quantum harmonic oscillator, you often encounter expressions involving square roots. Simplifying these expressions makes it easier to understand the relationships between different variables and perform further calculations. Similarly, in engineering, simplifying radicals can help in optimizing designs and reducing computational complexity.
