Problem 22 - Prealgebra

Question

Simplify each of the following expressions without using a calculator. $$6 \sqrt{64}+11 \sqrt{100}$$

Step-by-Step Solution

Answer

The simplified expression is 158.

Step 1: Identify each square root

The expression given is \(6 \sqrt{64} + 11 \sqrt{100}\). First, identify the square roots: \(\sqrt{64}\) and \(\sqrt{100}\).

Step 2: Simplify the square roots

Calculate the square roots: \(\sqrt{64} = 8\) and \(\sqrt{100} = 10\).

Step 3: Multiply by the coefficients

Multiply each square root by its corresponding coefficient: \(6 \times 8 = 48\) and \(11 \times 10 = 110\).

Step 4: Add the results

Add the two results from Step 3: \(48 + 110 = 158\).

Key Concepts

Square Roots

Square roots are a fundamental part of algebra and mathematics in general. They help us find the value which, when multiplied by itself, gives the original number. In simpler words, the square root of a number \(x\) is a number \(y\) such that \(y \times y = x\). For example, the square root of 64, written as \(\sqrt{64}\), is 8 because \(8 \times 8 = 64\). Similarly, \(\sqrt{100}\) is 10, since \(10 \times 10 = 100\).

• To find a square root, look for a number that squares to the original.
• Square roots can be perfect, like 64 or 100, meaning they have whole number results.
• Square roots can also be irrational, with non-repeating, non-terminating decimal values, like \(\sqrt{2}\).

Understanding square roots is crucial in simplifying mathematical expressions, as seen in our original exercise.

Coefficients

Coefficients are the numerical part of terms that contain variables or, as in our case, expressions with roots. In the expression \(6 \sqrt{64} + 11 \sqrt{100}\), the numbers 6 and 11 are coefficients. They show how many times the square root is taken into account in that term.

• Coefficients multiply the variable or the number by that factor.
• In expressions with roots, coefficients help maintain the balance by scaling the result of the root operation.
• In our example, the result of \(6 \times \sqrt{64}\) gives more weight to the term involving 64.

Understanding coefficients helps you organize and simplify expressions by revealing the structure of each term.

Addition of Terms

The addition of terms involves summing up values to create a simpler expression. It's important to ensure that the terms are simplified to their most basic form first before adding them together. Once the square roots were simplified in our example and multiplied by their coefficients, we were left with two numbers: 48 and 110. The final step involves adding these two numbers together.

• Make sure each term in the expression is fully simplified before performing addition.
• This step involves basic arithmetic; however, ensure accuracy in calculations as errors here can affect the final result.
• Once simplified terms are established, sum them up to get the final simplified expression.

By understanding how to add terms, you can simplify even complex expressions efficiently.

Mathematical Operations

Mathematical operations are the building blocks of simplifying expressions and solving equations. These operations include addition, subtraction, multiplication, and division. When simplifying expressions such as \(6 \sqrt{64} + 11 \sqrt{100}\), you need to execute these operations step-by-step in the correct order to achieve the correct result.

• Identify the operations involved: multiplication and addition in our example.
• Follow the order of operations - if there are parentheses, then exponents, then multiplication/division, and finally addition/subtraction (PEMDAS/BODMAS).
• In our solution, we multiplied first and then added the results, following the proper sequence.

Being familiar with mathematical operations is essential for simplifying expressions and solving mathematical problems accurately.