Problem 11 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is 48.

Step 1: Identify the square root

The expression given is \(6 \sqrt{64}\). The first task is to identify and simplify the square root. We need to find the square root of 64, which is a perfect square. We know that \(8 \times 8 = 64\), hence \(\sqrt{64} = 8\).

Step 2: Rewrite the expression

Now that we have found \(\sqrt{64} = 8\), we rewrite the original expression by substituting 8 in place of \(\sqrt{64}\). This changes the expression to \(6 \times 8\).

Step 3: Perform the multiplication

Finally, we need to multiply the numbers remaining in the expression. This step involves calculating \(6 \times 8 = 48\).

Key Concepts

Square Roots

Square roots are mathematical operations that undo the effect of squaring a number. Essentially, the square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 64 is 8 because \(8 \times 8 = 64\). A square root is often represented by the radical symbol \(\sqrt{}\).

• If a number is a perfect square, like 64, its square root is an integer.
• Finding square roots involves knowing which numbers squared (or multiplied by themselves) produce a given number.

Recognizing perfect squares helps simplify expressions quickly because it allows you to replace the square root with its corresponding whole number. When simplifying, always check if the number is a perfect square.

Multiplication

Multiplication is one of the basic arithmetic operations, crucial when simplifying expressions involving square roots and other numbers. In our exercise, once the square root of 64 is simplified to 8, the next task is to multiply it by 6.Multiplication is essentially repeated addition. For instance, \(6 \times 8\) means adding the number 6 eight times, or the number 8 six times, resulting in 48.

• Always ensure you perform multiplication steps after handling other operations like square roots.
• Use multiplication to combine numbers into a single simpler expression.

Understanding multiplication as grouping or repeated addition can help in visualizing and verifying your calculation. Breaking down expressions into simple components aids in the overall simplification process.

Perfect Squares

Perfect squares are numbers that result from an integer multiplied by itself. These are crucial in simplifying square roots, as identifying them immediately gives you the value of the square root. In the exercise, 64 is identified as a perfect square since \(8 \times 8 = 64\).Recognizing perfect squares can simplify mathematical problems significantly. Here are few examples:

• \(1^2 = 1\)
• \(2^2 = 4\)
• \(3^2 = 9\)
• \(4^2 = 16\)
• \(5^2 = 25\)

When you recognize numbers like 64, 81, or 100 as perfect squares, it saves time in calculation. Using these efficiently in problems lets you bypass lengthy arithmetic and move directly to simplified results.