Problem 14 - Intermediate Algebra

Question

Use radical notation to write each expression. Simplify if possible. $$ -64^{1 / 2} $$

Step-by-Step Solution

Answer

The simplified expression is \(-8\).

Step 1: Understand the Expression

The given expression is \[-64^{1/2}\]. This is a radical expression (square root) because of the exponent \( 1/2 \), which signifies taking the square root of the base.

Step 2: Apply Radical Notation

Rewrite the expression using radical notation:\[-64^{1/2} = -\sqrt{64}\]The negative sign is outside the square root in this expression.

Step 3: Simplify the Radical

Calculate the square root of 64, which is 8, as 64 is a perfect square:\[\sqrt{64} = 8\]

Step 4: Apply the Negative Sign

Since the negative sign is outside the radical, apply it to the result:\[-\sqrt{64} = -8\]

Key Concepts

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. It's like asking, "What number times itself equals this number?" The square root is commonly represented by the radical symbol \(\sqrt{}\). If you see an expression like \(\sqrt{64}\), it's asking you to find the number that, when squared, equals 64. In this case, it's 8, because \(8 \times 8 = 64\). In math, dealing with square roots can sometimes be a bit tricky because not all numbers have a neat, whole number as a square root. However, when they do, we say the number is a perfect square.A few key points about square roots:

• The square root of a positive number is always positive.
• Every positive number has two square roots, one positive and one negative (e.g., \(\sqrt{36} = 6\) and \(-\sqrt{36} = -6\)).
• The square root of zero is zero.

Radical Expressions

Radical expressions are expressions that contain a square root, cube root, or other roots. In the world of radicals, we often work with the square root the most. A radical expression can look like \(-\sqrt{64}\) or even something more complex like \(\sqrt{x^2 + 4}\). When you have \(64^{1/2}\), you can convert it to radical notation: this is equivalent to \(\sqrt{64}\).Radical expressions can be simplified while preserving their equivalence. Simplifying radicals often involves factoring the expression inside the radical symbol to find a perfect square factor. Once identified, the square root of the perfect square can be taken out from the radical.

• A negative sign outside the radical symbolizes that the entire value is negative after the operation.
• Simplified radical expressions make it easier to work with them in further algebraic operations.

Understanding radical notation is crucial for working with algebraic expressions and is fundamental in higher-level math studies.

Perfect Squares

Perfect squares are numbers that are the square of an integer. In simpler terms, if you multiply a number by itself and get an integer result, that result is a perfect square. For instance, \(36\) is a perfect square because it equals \(6 \times 6\). In the context of the problem, \(64\) is a perfect square because \(8 \times 8 = 64\).Recognizing perfect squares is helpful in simplifying square roots, as seen when the expression \(\sqrt{64}\) was simplified to \(8\). This not only simplifies calculations but also helps in recognizing and working with more complex radical expressions.A look at some useful perfect squares:

• \(1 = 1 \times 1\)
• \(4 = 2 \times 2\)
• \(9 = 3 \times 3\)
• \(16 = 4 \times 4\)
• \(25 = 5 \times 5\)
• ... and so on.

Once you've memorized or become familiar with these numbers, simplifying radicals becomes much quicker and easier. It’s all about recognizing that familiar multiplication that makes a number a perfect square.