Problem 106 - Elementary and Intermediate Algebra

Question

Look Alikes \(\cdots\) a. \(\sqrt{2}+\sqrt{18}\) b. \(\sqrt{2}+\sqrt{19}\)

Step-by-Step Solution

Answer

(a) simplifies to \(4\sqrt{2}\); (b) remains \(\sqrt{2} + \sqrt{19}\), so they are not alike.

Step 1: Simplify Square Roots

First, let's simplify the square roots where possible. For part (a), we can simplify \( \sqrt{18} \):\[ \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \]So the expression becomes:\( \sqrt{2} + 3\sqrt{2} = 4\sqrt{2} \)For part (b), \( \sqrt{19} \) is already in its simplest form as 19 is a prime number, so the expression remains:\( \sqrt{2} + \sqrt{19} \)

Step 2: Add Like Terms

In part (a), we can add the like terms:\( \sqrt{2} + 3\sqrt{2} = 4\sqrt{2} \)In part (b), there are no like terms to combine, so we keep:\( \sqrt{2} + \sqrt{19} \).

Step 3: Evaluate for Similarity

Notice that in (a), we have combined the terms to get \( 4\sqrt{2} \), which is a single term with \( \sqrt{2} \).In part (b), the terms \( \sqrt{2} \) and \( \sqrt{19} \) are not like terms, hence they cannot be simplified or combined further. Thus, part (a) results in a simpler single-term expression, whereas part (b) remains a two-term expression, showing they are not alike in structure.

Key Concepts

Prime Numbers

A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. In simpler terms, a prime number is only divisible by 1 and itself without leaving a remainder.
For example, consider the number 19. You can't divide it evenly by any number other than 1 and 19. This is why 19 is a prime number.

• All non-prime numbers greater than 1 are called composite numbers.
• The smallest prime number is 2, which is also the only even prime number.

When dealing with square roots, recognizing whether a number is prime can help determine if the expression can be simplified further. For instance, in the exercise, 19 is a prime number; thus, \( \sqrt{19} \) is left in its simplest form.

Like Terms

In algebra, like terms refer to terms that have the same variable raised to the same power. It doesn't matter what the coefficient is, as long as the variable and its exponent match, they can be combined.
For example, in the expression \( 3\sqrt{2} + \sqrt{2} \), both terms contain the square root of 2, or \( \sqrt{2} \), making them like terms. This is why they can be combined to \( 4\sqrt{2} \).

• Like terms have exactly the same variable factors.
• Combining like terms simplifies algebraic expressions and makes them easier to work with.

Identifying and combining like terms is a crucial skill in algebra, facilitating the simplification process and leading to expressions that are easier to understand and solve.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operational symbols. It represents a particular value or set of values.
A simple example is the expression \( x + 2 \), where "x" can represent any number. Solving the expression involves finding what value of "x" makes the equation true.

• Expressions can vary from simple ones, such as \( x + 2 \), to more complex ones like \( \sqrt{x} + y^2 \).
• They often contain constants (fixed values), variables (symbols representing numbers), and coefficients (numbers multiplied by variables).

In the given exercise, both parts (a) and (b) are algebraic expressions involving square root elements, which we simplify by combining like terms or recognizing that they are already in their simplest form.