Problem 58 - Algebra 1: Concepts and Skills

Question

Which of the following is equal to the difference \(\sqrt{3}-5 \sqrt{9} ?\) $$ f.\quad \sqrt{3}-15 $$ $$ g.\quad -4 \sqrt{3} $$ $$ h.\quad \sqrt{3}-3 $$ $$ j.\quad 3+2 \sqrt{5} $$

Step-by-Step Solution

Answer

The correct answer is: f. \( \sqrt{3} - 15 \)

Step 1: Understanding the Exercise

The problem is to find out which of the options is equal to \( \sqrt{3} - 5\sqrt{9} \) . The process involves simplifying the expression to determine the correct match from the given choices.

Step 2: Simplifying \(\sqrt{3} \)

The square root of 3 is a number that when squared gives 3. It cannot be simplified any further because 3 is a prime number. So, \(\sqrt{3} \) stays as it is.

Step 3: Simplifying \( 5\sqrt{9} \)

The square root of 9 is 3. So, the term simplifies to \( 5 * 3 = 15 \).

Step 4: Performing the Subtraction

Subtract the simplified second term from the first term: \( \sqrt{3} - 15 \)

Key Concepts

Square Roots

Understanding square roots can be both fascinating and very helpful in simplifying algebraic expressions. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because when you multiply 3 by 3, you get 9.
In algebra, square roots can pop up frequently, especially with non-perfect squares, like 3. Since there's no whole number that you can square to get 3, \( \sqrt{3} \) is left in its simplest form. It represents an irrational number—a number that cannot be written as a simple fraction. In contrast, when simplifying \(\(\sqrt{9}\)\\), we know this equals 3, because it is a perfect square.
Need some tips on dealing with square roots?- Identify whether the number is a perfect square.- For non-perfect squares, leave them in square root form unless told otherwise.- If you have a coefficient in front of a square root, multiply that with the simplified square root (like in the formula \( 5\sqrt{9} = 15 \)).

Algebraic Subtraction

When working with algebraic subtraction, especially involving square roots, some rules can simplify the process. Subtraction of numbers and expressions in algebra follow the same basic principles as regular arithmetic subtraction. However, when working with square roots, it's important to handle them with care.
Consider the expression \(\(\sqrt{3} - 5\sqrt{9}\)\). Here, it helps to simplify any square roots first, like we did with \(5\sqrt{9} = 15\). Now, you subtract the numbers as you'd do in basic subtraction, catching that \(\(\sqrt{3} - 15\)\) gives the rewritten solution. Remember: only like terms, such as two similar square roots, can be combined directly. Unalike terms must remain in their respective forms, creating clear and concise results.
Useful pointers for algebraic subtraction:- Simplify square root terms before subtracting.- Combine only like terms.- Keep the negative sign connected to the term it belongs to for accuracy.

Prime Numbers

Prime numbers play a key role in both basic arithmetic and algebra. A prime number is defined as a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. Examples include 2, 3, 5, 7, and so on. These numbers have exactly two distinct positive divisors: 1 and themselves.
Learning to identify prime numbers helps when simplifying square roots. For instance, knowing that 3 is a prime number, we recognize that \( \sqrt{3} \) cannot be simplified, because no smaller pair of whole numbers can produce it through multiplication. This understanding is crucial in algebra, where simplifying expressions like \( \sqrt{3} + \text{other terms} \) depends on the component number properties.
Why are prime numbers essential in algebra?- They indicate the simplest form of a square root when the number isn't a perfect square.- They help in determining factorization of numbers in expressions.- Being aware of prime numbers ensures accuracy in algebraic operations involving roots.