Question

\(\mathrm{R}=(3+\sqrt{5})^{2 \mathrm{n}}\) and \(\mathrm{f}=\mathrm{R}-[\mathrm{R}]\), Where [] is an integer part function then \(\mathrm{R}(1-\mathrm{f})=\) (a) \(2^{2 \mathrm{n}}\) (b) \(4^{2 \mathrm{n}}\) (c) \(8^{2 \mathrm{n}}\) (d) \(1^{2 \mathrm{n}}\)

Short answer

The correct answer for \(\mathrm{R}(1-\mathrm{f})\) is (d) \(1^{2 \mathrm{n}}\).

Step-by-step solution

Express f as a subtraction between R and integer part of R

We are given the expression for \(\mathrm{f}=\mathrm{R}-[\mathrm{R}]\) where \([\mathrm{R}]\) represents the integer part of \(\mathrm{R}\). We will use this expression later when computing \(\mathrm{R}(1-\mathrm{f})\).

Find 1-f

We need to compute \(1-\mathrm{f}\). Using the expression for \(\mathrm{f}\) we found earlier, we can rewrite this as: \(1-\mathrm{f} = 1 - (\mathrm{R}-[\mathrm{R}])\). We can then simplify this as: \(1-\mathrm{f}=1-\mathrm{R}+[\mathrm{R}]=[\mathrm{R}]-\mathrm{R}+1\)

Calculate R(1-f)

Now that we have the expression for \(1-\mathrm{f}\), we can compute \(\mathrm{R}(1-\mathrm{f})\). We can substitute the expression for \([\mathrm{R}] - \mathrm{R} + 1\) that we found in Step 2: \(\mathrm{R}(1-\mathrm{f})=\mathrm{R}([\mathrm{R}]-\mathrm{R}+1)\) \((3+\sqrt{5})^{2n}([\mathrm{R}]-\mathrm{R}+1)\)

Notice the Binomial Theorem Pattern

Since \(\mathrm{R}\) has the form of \((3+\sqrt{5})^{2n}\), it will expand using the binomial theorem. Pay attention to the even powers of the expansion, as odd powers will include irrational numbers. We want to understand the relationship between the integer part of \(\mathrm{R}\) and the irrational part of \(\mathrm{R}\). Consider the expansion of \((3+\sqrt{5})^2\). The even terms are: \((3+\sqrt{5})^2= 3^2 + 2(3)(\sqrt{5}) + 5= 9 + 6\sqrt{5} + 5= 14 + 6\sqrt{5}\). We can conclude that the integer part of \((3+\sqrt{5})^2\) is 14 and the irrational part is \(6\sqrt{5}\).

Compute R(1-f) and Match the Answer

From the reasoning in Step 4, we can see that \(\mathrm{R}\) will generally have terms containing even powers of 3 and 5 (which will be integer) and the irrational parts containing some multiples of \(\sqrt{5}\). Thus, the term \([\mathrm{R}]-\mathrm{R} +1\) will have an irrational part that cancels out the irrational part of \(\mathrm{R}\). Thus, when looking at the expanded form of \(\mathrm{R}(1-\mathrm{f})\), only the even powers of the expansion and the integer coefficients matter. Since the constant term in the expansion is 1, our \(\mathrm{R}(1-\mathrm{f})\) has the form of \(1^{2n}\). Thus, the correct answer is: (d) \(1^{2 \mathrm{n}}\)

Key concepts

Binomial Theorem

The Binomial Theorem is a powerful tool used to expand expressions that are raised to a power, such as \((x+y)^n\). It provides a formula that allows for the expression to be expanded into a sum involving terms with different combinations of the variables \(x\) and \(y\).
The formula is:

• \( (x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \)

In simpler terms, it breaks down the power into multiple terms with specific coefficients \(\binom{n}{k}\), representing the number of ways to choose \(k\) elements from \(n\). These terms include both integer terms (even powers) and terms that may involve irrational numbers (odd powers when irrational numbers are present).
For our problem, \((3+\sqrt{5})^{2n}\) uses the Binomial Theorem, leading to both integer components and irrational components based on the power of \(\sqrt{5}\). The integer part involves even powers, while the irrational part involves odd powers.

Irrational Numbers

An irrational number cannot be expressed as a simple fraction or a ratio of two integers. Common examples include \(\sqrt{2}\), \(\pi\), and \(e\).
Irrational numbers have non-repeating, non-terminating decimal expansions which make them more complex to work with in calculations. In our exercise, \(\sqrt{5}\) is an irrational number.
When involving irrational numbers in expressions like \((3+\sqrt{5})^{2n}\), they appear in terms with odd powers in the binomial expansion. This characteristic separates the integer and irrational parts of expressions, which is crucial for understanding their complete structure. While integer parts are straightforward, irrational parts can influence the non-integer value of an expression.

Exponentiation

Exponentiation refers to the operation of raising a number to the power of another, represented as \(a^n\). This means multiplying the number \(a\) by itself \(n\) times.
Key points about exponentiation:

• Base: The number being multiplied. In \(2^3\), 2 is the base.
• Exponent: The power to which the base is raised. In \(2^3\), 3 is the exponent.
• Result: \(2^3 = 2 \times 2 \times 2 = 8\).

In our problem, the expression \((3+\sqrt{5})^{2n}\) exemplifies exponentiation where the base is \((3+\sqrt{5})\) and the exponent is \(2n\). Understanding the properties of exponentiation helps simplify complex mathematical expressions and solve equations like \(R(1-f)\).

Rational and Irrational Parts

In mathematical expressions, distinguishing between rational and irrational parts is essential.
- **Rational Numbers**: These can be expressed as fractions, such as \(\frac{1}{2}\), \(3\), or \(-0.75\). They have repeating or terminating decimal expansions.- **Irrational Numbers**: As mentioned, they are non-terminating, non-repeating decimals.\(\sqrt{5}\) is one example.
When dealing with expressions involving both types, like \((3+\sqrt{5})^{2n}\), separating them helps in analysis. The integer part \([\text{R}]\) comes from entirely rational components, while f accounts for the irrational portion as per \(f = R - [R]\).
Understanding these distinctions aids in practices like simplifying expressions and calculating results such as \(R(1-f)\) where irrational parts cancel out, leaving only the rational result.