Problem 3 - Algebra and Trigonometry

Question

Is there a difference between \(\sqrt{5^{2}}\) and \((\sqrt{5})^{2} ?\) Explain.

Step-by-Step Solution

Answer

No, both expressions simplify to 5, so there is no difference.

Step 1: Simplify \( \sqrt{5^{2}} \)

The expression \( \sqrt{5^{2}} \) involves taking the square root of \( 5 \) squared. We know that squaring and then taking the square root of the same number returns the number itself: \[ \sqrt{5^{2}} = \sqrt{25} = 5 \]. So, the result of this expression is 5.

Step 2: Simplify \( (\sqrt{5})^{2} \)

The expression \( (\sqrt{5})^{2} \) involves squaring the square root of \( 5 \). When you square a square root, the operations cancel out each other, leaving just the base: \[ (\sqrt{5})^{2} = 5 \]. Therefore, the result of this expression is also 5.

Step 3: Compare both expressions

In steps 1 and 2, we simplified both expressions and found that both \( \sqrt{5^{2}} \) and \( (\sqrt{5})^{2} \) equal 5. Hence, there is no difference between the two expressions when simplified.

Key Concepts

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 multiplied by 5 equals 25. The operation is denoted by the radical symbol \( \sqrt{} \). When dealing with algebraic expressions, the square root can significantly simplify or transform an expression.
To properly handle square roots in expressions, follow these steps:

• First, ensure you understand what number you are taking the square root of.
• Simplify inside the radical symbol first, if possible.
• Ensure accuracy by checking that the resultant number, when squared, returns the number under the radical.

Understanding how square roots interact with exponents is crucial, as they can seem daunting at first. However, remembering that they are simply the reverse operation of squaring can make things easier.

Exponents

Exponents are used to express repeated multiplication of a number by itself. For instance, \( 5^2 \) denotes \( 5 \times 5 \), which equals 25. The number 5 is referred to as the base, while 2 is the exponent. This representation simplifies calculation and allows us to present large numbers more concisely.
Here are some key points about exponents:

• Any number raised to the power of 2 is squared.
• An exponent of 0 returns 1, regardless of the base (except when the base is 0).
• Exponents and square roots are inverse operations.

Connecting exponents to the square root, when you square \( \sqrt{5} \), you are essentially undoing the square root, thus returning to the base of 5. This principle helps in simplifying calculations and understanding algebraic expressions.

Simplification

Simplification is a process of reducing an expression to its most concise form, without changing its value. By employing simplification, complex expressions become more manageable and easier to understand.
Steps to simplify expressions involving square roots and exponents include:

• Identify operations like squaring or taking square roots and address them first.
• Use inverse operations to cancel out certain steps, such as squaring and square roots.
• Ensure that all operations follow the correct order, especially within parentheses.

For example, both \( \sqrt{5^2} \) and \( (\sqrt{5})^2 \) simplify to 5, despite being written differently. This showcases the power of simplification in revealing the underlying simplicity of seemingly different expressions.