(b) Express
$\begin{array}{l}
3 \sqrt{3}+2 \sqrt{3} \\
2 \sqrt{5}-5 \sqrt{2}
$\end{array}
in the form $q+r \sqrt{5}$ where $g, f$ and $S$ are rational numbers
Answer (1)
Simplify the first expression: 3 3 + 2 3 = 5 3 .
Attempt to express 5 3 in the form q + r 5 , which is not possible.
Simplify the second expression: 2 5 − 5 2 .
Attempt to express 2 5 − 5 2 in the form q + r 5 , which is not possible. Therefore, the answer is that neither expression can be written in the specified form. Not possible .
Explanation
Simplifying the first expression We are asked to express $3
\sqrt{3}+2 \sqrt{3}$ and 2 5 − 5 2 in the form q + r 5 where q and r are rational numbers. Let's start by simplifying the first expression.
Expressing in the desired form The first expression is 3 3 + 2 3 . We can combine these terms since they both contain 3 .
3 3 + 2 3 = ( 3 + 2 ) 3 = 5 3 Now, we want to express 5 3 in the form q + r 5 , where q and r are rational numbers. We can write 5 3 = q + r 5 . However, since 3 and 5 are linearly independent over the rational numbers, the only way to express 5 3 in the form q + r 5 is if q = 0 and r = 0 . This is not possible. Therefore, 5 3 cannot be expressed in the form q + r 5 where q and r are rational numbers.
Simplifying the second expression Now let's consider the second expression, 2 5 − 5 2 . We want to express this in the form q + r 5 , where q and r are rational numbers. We can write 2 5 − 5 2 = q + r 5 .
Rearranging the equation, we have − 5 2 = q + ( r − 2 ) 5 . Since 2 and 5 are linearly independent over the rational numbers, the only way to satisfy this equation is if both sides are equal to zero. This means q = 0 and r − 2 = 0 , which implies r = 2 . However, this would mean − 5 2 = 0 , which is not true. Therefore, 2 5 − 5 2 cannot be expressed in the form q + r 5 where q and r are rational numbers.
Final Answer Therefore, neither of the given expressions can be expressed in the form q + r 5 where q and r are rational numbers.
Examples
This problem demonstrates how to manipulate and simplify expressions involving square roots. Understanding how to express numbers in different forms is crucial in various fields, such as physics and engineering, where you might need to simplify complex equations or approximate values for practical calculations. For example, when dealing with electrical circuits, you might encounter expressions involving square roots when calculating impedance or resonance frequencies. Being able to simplify and manipulate these expressions allows for easier analysis and design of the circuits.
