Problem 74 - Algebra 2

Question

Find each power. $$ (\sqrt{x}+1)^{2} $$

Step-by-Step Solution

Answer

\((\sqrt{x} + 1)^{2} = x + 2\sqrt{x} + 1.\)

Step 1: Identify the expression

The expression we need to find the power of is \((\sqrt{x}+1)^{2}\). This means we need to square the binomial \(\sqrt{x} + 1\).

Step 2: Apply the Binomial Theorem

We use the formula for squaring a binomial: \((a + b)^{2} = a^{2} + 2ab + b^{2}\), where \(a = \sqrt{x}\) and \(b = 1\).

Step 3: Calculate \(a^{2}\)

Calculate \(a^{2}\) by squaring the first term. For \(a = \sqrt{x}\), we have \((\sqrt{x})^{2} = x\).

Step 4: Calculate \(2ab\)

Calculate \(2ab\), the cross term. Here, \(2ab = 2(\sqrt{x})(1) = 2\sqrt{x}\).

Step 5: Calculate \(b^{2}\)

Calculate \(b^{2}\) by squaring the second term. For \(b = 1\), we have \(1^{2} = 1\).

Step 6: Combine results

Combine the results from Steps 3, 4, and 5 to get the final expression: \((\sqrt{x} + 1)^{2} = x + 2\sqrt{x} + 1\).

Key Concepts

Algebraic Expressions

Algebraic expressions are fundamental in mathematics. They consist of variables, numbers, and operations like addition and multiplication. Understanding how to work with them is essential for solving many kinds of problems. In our exercise, the expression we analyze is \((\sqrt{x}+1)^2\). This expression is composed of:

• \(\sqrt{x}\), which is a square root expression involving a variable \(x\).
• The number \(1\), which is a constant added to the square root term.

Algebraic expressions often require simplification by combining like terms and applying various algebraic identities, such as the binomial theorem, to solve problems or make calculations easier.
This ability to manipulate and simplify expressions is critical in mathematics, as it lays the groundwork for more advanced topics like calculus and trigonometry.

Exponentiation

Exponentiation is a mathematical operation where a number, known as the base, is raised to a power, which is the exponent. In our case, the expression is \((\sqrt{x}+1)^{2}\). Here, we are raised the base \((\sqrt{x}+1)\) to the power of \(2\).
The binomial theorem provides a handy formula for expanding binomial expressions that are raised to a power. According to this theorem, \((a+b)^2 = a^2 + 2ab + b^2\). This formula helps break down the problem into manageable pieces and find individual terms.
Steps involved:

• Square the first term \(a\), which in this case is \(\sqrt{x}\).
• Multiply the first and second terms together and double the result.
• Square the last term \(b\), which is \(1\) here.

By mastering exponentiation and applying the binomial theorem, you can simplify expressions and solve equations efficiently.

Square Roots

Square roots are inverses of squaring a number. For example, if \(x^2 = y\), then \(\sqrt{y} = x\). In our expression, we deal with \(\sqrt{x}\), which signifies a number that, when squared, returns \(x\).
Square roots appear frequently in mathematics and have particular properties which are vital for simplification. When combined with variables in algebraic expressions, they need careful handling to ensure the correct simplification.
Key points involved:

• A square root will always return the principal (positive) root in algebraic contexts, unless specified otherwise.
• Simplifying expressions involves properly applying square root rules, like \(\sqrt{a^2} = a\), wherever needed.
  
• Ensuring the expression remains valid, especially when variables, like \(x\), could represent any number, is crucial.

Understanding and using square roots in expressions helps solve not only textbook exercises but also real-world problems involving measurements, geometry, and physics.