Problem 106 - Intermediate Algebra

Question

Evaluate each expression. $$ \text { a. }(-2-\sqrt{64})^{2} \quad \text { b. }-2-(\sqrt{64})^{2} $$

Step-by-Step Solution

Answer

Part a evaluates to 100, and part b evaluates to -66.

Step 1: Simplify the square root

First, calculate the square root of 64. Since 64 is a perfect square, we have \( \sqrt{64} = 8 \).

Step 2: Substitute and evaluate part a

Now, substitute 8 into the expression for part a: \((-2-\sqrt{64})^2\). This simplifies to \((-2-8)^2 = (-10)^2\).

Step 3: Compute the square

Find the square of -10. \((-10)^2 = 100\). Thus, the simplified value for part a is 100.

Step 4: Substitute and evaluate part b

For part b, substitute 8 into the expression: \(-2-(\sqrt{64})^2\). This becomes \(-2 - 8^2\).

Step 5: Compute the power

Calculate \(8^2\): \(8^2 = 64\).

Step 6: Simplify the expression

Combine the terms in the expression, \(-2 - 64 = -66\). Thus, the simplified value for part b is -66.

Key Concepts

Square Roots

When we talk about square roots, we're referring to a mathematical operation that finds a number which, when multiplied by itself, gives the original number. For example, the square root of 64 is 8. This is because 8 times 8 equals 64. Square roots are often represented using the radical symbol, \( \sqrt{} \).

• To find the square root of a number, you can think of it as reversing the process of squaring that number.
• If a number is a perfect square, its square root will be an integer.

Understanding square roots is crucial when evaluating expressions, as it helps in simplifying terms or solving equations. In our exercise, knowing that \( \sqrt{64} = 8 \) allows us to substitute this value into the expressions, simplifying them for further operations.

Perfect Squares

A perfect square is a number that can be expressed as the product of an integer multiplied by itself. For instance, 64 is a perfect square because it equals \( 8 \times 8 \).

• Recognizing perfect squares helps in simplifying expressions and solving algebraic equations.
• Common perfect squares include numbers like 1, 4, 9, 16, 25, 36, 49, etc.

In mathematical expressions, identifying a perfect square can allow you to quickly find square roots or manipulate terms with confidence. For example, in the expression \((-2-\sqrt{64})^{2}\), by noticing that 64 is a perfect square, you can simplify the square root operation to make the further evaluation straightforward.

Simplifying Expressions

Simplifying expressions is a key part of algebra that involves reducing expressions to their simplest form. This is done by performing operations such as addition, subtraction, multiplication, division, squaring, or taking square roots.

• Simplification often involves combining like terms or converting complex operations into simpler ones.
• For expressions involving brackets or powers, simplify step by step: address innermost operations first, follow order of operations (PEMDAS/BODMAS), and simplify as you go.

In the exercise provided, we demonstrated simplification by first evaluating the square root, then substituting it into the expressions, and finally performing arithmetic operations.
For instance, in part a, this involved turning the expression \((-2-8)^{2}\) into a more manageable \((-10)^{2}\), and finally \(100\). This step-by-step process ensures clarity and accuracy in solving the expressions.