Question

The number of terms which are not radicals in the expansion \((\sqrt{7}+4)^{6}+(\sqrt{7}-4)^{6}\), after simplification is (1) 6 (2) 5 (3) 4 (4) 3

Short answer

Answer: 3

Step-by-step solution

Expand the expressions using the binomial theorem

Recall the binomial theorem: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). Using this, we can expand both expressions: \((\sqrt{7}+4)^{6} = \sum_{k=0}^{6} \binom{6}{k} (\sqrt{7}^{6-k}) (4^k)\) \((\sqrt{7}-4)^{6} = \sum_{k=0}^{6} \binom{6}{k} (\sqrt{7}^{6-k}) ((-4)^k)\) Now, we have to find the sum of both expansions: \((\sqrt{7}+4)^{6}+(\sqrt{7}-4)^{6} = \sum_{k=0}^{6} \binom{6}{k} (\sqrt{7}^{6-k}) (4^k) + \sum_{k=0}^{6} \binom{6}{k} (\sqrt{7}^{6-k}) ((-4)^k)\)

Simplify the sum and identify terms with radicals

Combine the two sums into one sum: \(= \sum_{k=0}^{6} \binom{6}{k} (\sqrt{7}^{6-k}) (4^k+(-4)^k)\) Observe that for even k, \(4^k+(-4)^k = 2\cdot4^k\) and for odd k, \(4^k+(-4)^k = 0\). Furthermore, \(\sqrt{7}^{6-k}\) is not a radical if and only if \(6-k\) is even.

Count the terms without radicals

To count the terms without radicals, we must look for even values of k that will give even values of 6-k, which are k = 0, 2, and 4. For k = 0, the term is \(\binom{6}{0} (\sqrt{7}^6) (2\cdot4^0) = \sqrt{7}^6\) (Not a radical) For k = 2, the term is \(\binom{6}{2} (\sqrt{7}^4) (2\cdot4^2) = 15\cdot7^2\cdot32\) (Not a radical) For k = 4, the term is \(\binom{6}{4} (\sqrt{7}^2) (2\cdot4^4) = 15\cdot7\cdot2048\) (Not a radical) No other terms meet the criteria. Therefore, there are 3 terms that are not radicals in the simplified expression. The correct answer is (4) 3.

Key concepts

Radicals

Understanding radicals is essential when dealing with expressions like \((\sqrt{7}+4)^6\). A radical fundamentally involves the root of a number, such as \(\sqrt{7}\), which means the square root of 7. Radicals are often irrational numbers unless the number under the root is a perfect square.
When solving problems, avoiding or simplifying radicals can make calculations much easier. In the context of polynomial expansion, whether a term represents a radical or not depends on if the base of the root (like 7 in \(\sqrt{7}\)) becomes raised to an integer that eliminates the square root. For instance, \((\sqrt{7})^2 = 7\) removes the radical because the square and the square root are inverse operations.
In our exercise, we're searching for terms in a binomial expansion that do not retain a radical. This occurs only when the power to which the radical component is raised results in a whole number, making the term not a radical.

Polynomial Expansion

The expansion of a polynomial such as \((\sqrt{7}+4)^6\) leverages the binomial theorem. This theorem helps to expand expressions of the form \((a + b)^n\) into sums of terms which involve coefficients, powers of \(a\), and powers of \(b\).

• The binomial theorem states: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\).
• Here, \(\binom{n}{k}\) is a binomial coefficient, representing the number of ways to choose \(k\) elements from \(n\).
• This expansion offers terms that can be manipulated to find either simplified forms or specific coefficients, like the terms containing radicals.

By expanding both \((\sqrt{7}+4)^6\) and \((\sqrt{7}-4)^6\), each term is examined for the presence of radicals. Only terms with an integer result in removing the radical component which stems from one of the binomial terms not having a pesky square root intact.

Mathematical Simplification

Simplification is a vital process in solving equations or expressions that involve complex terms, often including radicals or many components that can influence the solvability of a problem.
When we simplify the expression \((\sqrt{7}+4)^6 + (\sqrt{7}-4)^6\), we look for opportunities to reduce the complexity by eliminating radicals, combining like terms, or resolving expressions into fewer terms.

In the step-by-step solution, using symmetry and properties of numbers, we simplify

• The sums like \(4^k + (-4)^k\), which resolve to 0 for odd powers and provide tangible results for even powers.
• The examination of individual terms of the expansion to determine whether they retain a radical or not.

Applying these techniques results in an expression that is easier to manage, interpret, and eventually solve, providing clearer insights, like identifying how many non-radical terms remain. Through simplification, the seemingly complex expansion is distilled into its essence, allowing for effective evaluation and understanding.