Problem 17 - Prealgebra

Question

Simplify each of the following expressions without using a calculator. $$\sqrt{49}+\sqrt{64}$$

Step-by-Step Solution

Answer

15

Step 1: Identify the Square Root

Determine the square roots of the numbers inside the radicals. For \( \sqrt{49} \), we need to find a number \( x \) such that \( x^2 = 49 \).

Step 2: Calculate \( \sqrt{49} \)

Since \( 7^2 = 49 \), we know \( \sqrt{49} = 7 \).

Step 3: Calculate \( \sqrt{64} \)

Similarly, find a number \( y \) such that \( y^2 = 64 \). Since \( 8^2 = 64 \), \( \sqrt{64} = 8 \).

Step 4: Add the Results Together

Now that we know \( \sqrt{49} = 7 \) and \( \sqrt{64} = 8 \), we can add them together: \( 7 + 8 = 15 \).

Key Concepts

Square Roots

A square root is a fascinating mathematical operation that essentially asks the question: "What number multiplied by itself will give the original number?" This operation is represented by the radical symbol \( \sqrt{} \). For example, when we see \( \sqrt{49} \), we are looking for a number which, when squared, results in 49.
To solve this, identify a number like 7 because \( 7 \times 7 = 49 \). Therefore, \( \sqrt{49} = 7 \).

• Square roots apply only to non-negative numbers in most situations, as squares of negative numbers result in positive outcomes.
• Finding the square root of a perfect square is usually straightforward because perfect squares have whole numbers as their square roots.

As you explore square roots, remember this operation is the inverse of squaring a number. It plays a crucial role in various mathematical concepts and real-life applications, from geometry to physics.

Radicals

Radicals, represented by the symbol \( \sqrt{} \), extend beyond just square roots. They actually describe roots of various degrees, such as cube roots (\( \sqrt[3]{x} \)) and beyond. However, the most common radical is the square root.
In essence, radicals allow us to work with roots in an algebraic manner, treating them like variables in equations or simplifying expressions. When simplifying expressions involving radicals, it is essential to understand how to manipulate and combine them.

• Radicals follow specific rules, such as \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \).
• Similarly, dividing radicals uses the rule \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \), provided \( b eq 0 \).

Radicals are useful because they represent solutions to polynomial equations that don't resolve neatly into integers or simple fractions. Getting comfortable with radicals is essential for solving more complex math problems later on.

Basic Arithmetic

Basic arithmetic forms the foundation of mathematics. It includes the primary operations you often encounter: addition, subtraction, multiplication, and division.
These operations are critical when simplifying expressions, as seen in the problem \( \sqrt{49} + \sqrt{64} \).
Here's how basic arithmetic ties into simplifying expressions:

• Addition is used to combine the outcomes once individual elements have been simplified.
• Subtraction, multiplication, and division perform similar roles for other types of expressions.

For example, once you find that \( \sqrt{49} = 7 \) and \( \sqrt{64} = 8 \), basic arithmetic lets you add these values: \( 7 + 8 = 15 \).
This part of mathematics is not only fundamental but also increasingly important as problems become more intricate. Strong arithmetic skills ensure accuracy and ease when approaching more advanced topics.