Problem 19 - Prealgebra

Question

Simplify each of the following expressions without using a calculator. $$\sqrt{16}-\sqrt{9}$$

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Identify Perfect Squares

Examine each part of the expression to recognize the perfect square numbers present. In the expression \( \sqrt{16} - \sqrt{9} \), the numbers 16 and 9 are both perfect squares.

Step 2: Simplify Each Square Root

Simplify each square root by calculating the square roots of the perfect squares identified. \( \sqrt{16} \) simplifies to 4 because 4 squared (\(4^2\)) equals 16. Similarly, \( \sqrt{9} \) simplifies to 3 because 3 squared (\(3^2\)) equals 9.

Step 3: Subtract the Resulting Numbers

Subtract the two simplified numbers obtained from the square roots. Perform the subtraction \( 4 - 3 \), which equals 1.

Key Concepts

Perfect Squares

In mathematics, a perfect square is a number that can be expressed as the product of an integer with itself. For example, 16 and 9 in our exercise are perfect squares because:

• 16 can be written as \(4 \times 4\), which means it is the square of 4.
• Similarly, 9 can be written as \(3 \times 3\), showing that it is the square of 3.

Recognizing these numbers helps us simplify expressions involving square roots, as you'll always look for integers when simplifying square roots of perfect squares. A common method is to check each number and see if the results from the product of smaller numbers match your perfect square.

Whenever you encounter the term "perfect square," remember:

• The square roots of perfect squares are integers.
• Identifying perfect squares lets you simplify expressions quickly.
• Perfect squares make calculations more straightforward by limiting the possible outcomes to whole numbers.

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, in our exercise:

• The square root of 16 is 4 because \(4 \times 4 = 16\).
• The square root of 9 is 3 because \(3 \times 3 = 9\).

In the context of simplifying radical expressions, calculating square roots is a crucial step. When the numbers under your radical sign are perfect squares, the square roots will always simplify to whole numbers.

It's important to note:

• Calculating square roots can involve estimating for non-perfect squares, but for perfect squares, we obtain exact answers rather easily.
• Being proficient with square roots allows for more efficient mathematical problem-solving, especially in algebraic manipulation.
• The inverse operation, squaring a number, can also aid in understanding and checking your results.

Subtraction of Rational Numbers

Subtraction is one of the basic arithmetic operations where you find the difference between numbers. In our exercise, we subtract two results obtained from the square roots, 4 and 3, to find the difference:

• Performing the subtraction \(4 - 3\) yields 1.

Rational numbers, like the integers involved here, are numbers that can be expressed as the quotient or fraction of two integers. Both 4 and 3 fall into this category, making this subtraction straightforward.

When dealing with subtraction of rational numbers, consider the following:

• Align numbers correctly to ensure accurate computation, especially with larger digits or decimals.
• Use subtraction as a tool to solve equations or simplify expressions.
• The concept of borrowing applies when subtracting larger digits from smaller ones, although it's unnecessary in our smaller integer example.

Getting familiar with subtraction aids not only in basic computation but also improves skills in handling more complex equations.