Problem 17 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ 7 \sqrt{9}-7+\sqrt{3} $$

Step-by-Step Solution

Answer

The expression simplifies to \(14 + \sqrt{3}\).

Step 1: Simplify Square Roots

First, simplify the square root present in the expression. The expression is \(7\sqrt{9} - 7 + \sqrt{3}\). Since \(\sqrt{9} = 3\), replace \(\sqrt{9}\) with \(3\).

Step 2: Multiply Coefficients

Multiply the outside coefficient by the simplified square root from Step 1. This means we have: \(7 \times 3 = 21\). The expression now becomes \(21 - 7 + \sqrt{3}\).

Step 3: Simplify the Expression

Combine like terms. Here, the constants \(21\) and \(-7\) can be combined. \(21 - 7 = 14\). So, the expression simplifies to \(14 + \sqrt{3}\).

Key Concepts

Understanding Real Numbers

Real numbers are all the numbers you can think of on the number line. They include all the integers, fractions, and decimals. A real number can be positive, zero, or negative.
Examples of real numbers are:

• 3, which is a positive integer
• -1.5, which is a negative decimal
• 0, which is neither positive nor negative
• and even irrational numbers like \( \sqrt{2} \)

When dealing with real numbers, you can perform operations like addition, subtraction, multiplication, and division. However, should always assume variables represent positive real numbers unless otherwise stated. In problems like the given exercise, dealing with square roots requires the assumption that variables represent non-negative real numbers to ensure the square roots are real, which simplifies the problem and avoids any undefined situations.

Simplifying Square Roots

Square root simplification is about making square roots as simple as possible. You simplify square roots by finding a number that, when multiplied by itself, gives the number under the square root.
For example, the square root of 9 is simplified because \( \sqrt{9} = 3 \), since 3 multiplied by itself gives 9.
Here's how to simplify square roots step by step:

• Identify if the number under the square root is a perfect square, like 4, 9, or 16.
• Find its square root, which is a number multiplied by itself to give the perfect square.
• Replace the square root in your expression with its simplified form.

In our exercise, simplifying \( \sqrt{9} = 3 \) helps in reducing the complexity of the math involved, making the expression easier to work with.

Expression Simplification

Expression simplification involves rewriting a math expression in its simplest form. This means performing all possible arithmetic operations and combining like terms.
In our exercise, after simplifying the square root, the expression is \( 7 \times 3 - 7 + \sqrt{3} = 21 - 7 + \sqrt{3} \). Next, you combine the constant numbers like 21 and -7.
For successful expression simplification, follow these tips:

• Simplify any square roots or operations within parentheses first.
• Perform any arithmetic operations such as multiplication and addition.
• Combine like terms, often constants that can be added or subtracted.

In the final step, the expression \( 21 - 7 + \sqrt{3} \) becomes \( 14 + \sqrt{3} \), which is the simplest form of the expression. This helps make calculations easier and more efficient to interpret.