Problem 257 - Elementary Algebra

Question

In the following exercises, simplify. a) \((\sqrt{11})^{2}\) (b) \((-\sqrt{21})^{2}\)

Step-by-Step Solution

Answer

a) 11, b) 21

Step 1: Simplify \((\sqrt{11})^{2}\)

To simplify \((\sqrt{11})^{2}\), recall the property of exponents that \(\sqrt{a}^{2} = a\). Applying this property, \((\sqrt{11})^{2} = 11 \).

Step 2: Simplify (\(-\sqrt{21})^{2}\)

To simplify \((\-\sqrt{21})^{2}\), recall the property of exponents that \( (a\cdot b)^2 = a^2\cdot b^2 \). Here, we have \(-1 \cdot \sqrt{21}\). Therefore, \((-\sqrt{21})^{2} = (-1)^{2} \cdot (\sqrt{21})^{2} = 1 \cdot 21 = 21\).

Key Concepts

Square Roots

Square roots are useful in simplifying expressions involving repeated multiplication. The square root of a number, say \(a\), is a value which, when multiplied by itself, gives back \(a\). For instance, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\). A square root is denoted with the radical symbol \(\sqrt{}\). The property of a square root raised to the power of 2 is vital in simplification. Notably, \( (\sqrt{a})^2 = a\). This property states that squaring a square root returns the original number. This concept is applied in the examples where \( (\sqrt{11})^2 = 11\) and \( (\sqrt{21})^2 = 21\).

Properties of Exponents

Exponents are a way to express repeated multiplication. For example, \(a^n\) means multiplying \(a\) by itself \(n\) times. Several properties of exponents make simplifying expressions easier:

• \(a^m \cdot a^n = a^{m+n} \) - To multiply powers with the same base, add the exponents.
• \(\frac{a^m}{a^n} = a^{m-n} \) - To divide powers with the same base, subtract the exponents.
• \((a^m)^n = a^{m\cdot n} \) - To raise a power to another power, multiply the exponents.
• \( (ab)^n = a^n \cdot b^n \)- To raise a product to a power, apply the exponent to each factor individually.

In our examples, \( (\sqrt{11})^2\) simplifies by the first property, and \( (-\sqrt{21})^2\) uses the last property. Thus, \( (-1\cdot\sqrt{21})^2 = (-1)^2\cdot(\sqrt{21})^2\).

Simplification Process

Simplifying mathematical expressions is about rewriting them in their simplest form. This often involves using basic principles and properties to reduce the complexity. Here's a general approach to simplify expressions with exponents or square roots:

• Identify and isolate parts of the expression that can be simplified directly.
• Apply properties of exponents or square roots accordingly.
• Combine like terms where possible.
• Consistently check each step for errors to ensure the accuracy of the simplification process.

In our exercise, we started by recognizing that \(\sqrt{a}^2 = a\). For \( (\sqrt{11})^2\), directly applying this rule gave us \(11\). For \((-\sqrt{21})^2\), breaking it into components \((-1)^2\cdot(\sqrt{21})^2)=1\cdot21 \) simplified the expression to 21. This process makes solving complex expressions manageable and effective.