Problem 46 - Prealgebra

Question

Use the following information. Squaring a number and finding the square root of a number are inverse operations. That is, one operation undoes the other operation. Use inverse operations to evaluate each expression. $$(\sqrt{64})^{2}$$

Step-by-Step Solution

Answer

The expression \((\sqrt{64})^2\) equals 64.

Step 1: Understand the Expression

The expression given is \((\sqrt{64})^2\). Notice that it involves a square root and a square. Recognize that these operations are inverses of each other.

Step 2: Simplify the Square Root

Calculate the square root of 64. Since 64 is a perfect square, \(\sqrt{64} = 8\). Write the expression now as \((8)^2\).

Step 3: Squaring the Result

Now compute the square of 8. \(8^2 = 64\).

Step 4: Conclude with the Inverse Operation

Understand that the operation of taking the square root and then squaring the result returns the original number (assuming non-negative roots), which confirms the operations cancel each other out.

Key Concepts

Squaring Numbers

Squaring a number means multiplying the number by itself. For example, if you take the number 3 and square it, you compute its square by performing the operation: 3 times 3, resulting in 9. When a number is squared, you can think of it as raising the number to the power of 2. This is written in mathematical terms as \( x^2 \), where \( x \) represents the number being squared.

• Example: \( 4^2 = 16 \), because 4 times 4 equals 16.
• Note: Squaring a negative number will always result in a positive number since multiplying two negative numbers together gives a positive product.

This operation is fundamental in algebra and geometry, and it's the simplest form of exponentiation. Understanding squares sets the stage for tackling more complex mathematical concepts.

Square Roots

Finding the square root of a number is the opposite operation to squaring. The square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \). Mathematically, the square root is represented by the radical symbol \( \sqrt{} \). For example, the square root of 16 is \( 4 \), because 4 times 4 is 16.

• Symbol: \( \sqrt{} \)
• Examples: \( \sqrt{25} = 5 \), \( \sqrt{81} = 9 \)
• Important: Square roots can be both positive and negative since \( (-4)^2 \) also equals 16. However, by convention, we consider the principal (non-negative) root.

Square roots are crucial in solving quadratic equations and in geometry problems involving right triangles and circles. They represent a significant aspect of the inverse relationship in mathematics.

Perfect Squares

A perfect square is an integer that is the square of another integer. In other words, a perfect square is the product of some integer with itself. Examples of perfect squares include 1, 4, 9, 16, and 25, which can be written as \( 1^2 \), \( 2^2 \), \( 3^2 \), \( 4^2 \), and \( 5^2 \), respectively.

• Examples: \( 36 \) is a perfect square because it equals \( 6^2 \).
• Attributes: Numbers like 49, which equals \( 7^2 \), are always positive.

Understanding perfect squares helps in estimating square roots and in performing various functions within algebra. Identifying whether a number is a perfect square can also simplify solving polynomials since it can indicate factorizability.