Problem 6 - Algebra A Combined Function

Question

Use radical notation to rewrite each expression. Simplify if possible. $$ \left(\frac{1}{64}\right)^{1 / 2} $$

Step-by-Step Solution

Answer

The expression in radical form is \( \sqrt{\frac{1}{64}} = \frac{1}{8} \).

Step 1: Understanding Radical Notation

To rewrite an expression using radical notation, we need to express it as a root. An expression with an exponent of \( \frac{1}{n} \) is equivalent to taking the \( n \)-th root of the base.

Step 2: Rewriting the Expression in Radical Form

The original expression \( \left(\frac{1}{64}\right)^{1/2} \) means we need to take the square root of \( \frac{1}{64} \). In radical form, this is written as \( \sqrt{\frac{1}{64}} \).

Step 3: Simplifying the Expression

To simplify \( \sqrt{\frac{1}{64}} \), we take the square root of the numerator and the denominator separately. So, \( \sqrt{1} = 1 \) and \( \sqrt{64} = 8 \). Thus, \( \sqrt{\frac{1}{64}} = \frac{1}{8} \).

Key Concepts

Square Roots

When we talk about square roots, we're looking at a specific type of radical used frequently in mathematics. The square root of a number is simply another number which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because if you multiply 3 by itself, you get 9.
In terms of notation, the square root is represented with a radical symbol: \( \sqrt{} \). However, there is something important to remember: every positive number actually has two square roots - a positive and a negative one. Still, when we say 'square root', we refer to the positive root, unless specified otherwise.
Let's apply this to the problem \( \left(\frac{1}{64}\right)^{1/2} \). To express it using radical notation, we can say \( \sqrt{\frac{1}{64}} \), meaning we seek a number that, when squared, results back to \( \frac{1}{64} \). The square root helps simplify expressions by providing a direct value or easier form.

Exponents

Exponents describe how many times a number, the base, is multiplied by itself. For instance, \( 2^3 = 2 \times 2 \times 2 = 8 \). They are a shorthand notation to make expressions more concise and readable.
In the case where the exponent is a fraction, such as \( \frac{1}{2} \), it implies a root operation instead of repeated multiplication. So in our exercise, rewriting \( \left(\frac{1}{64}\right)^{1/2} \) in radical form is understood as the square root operation \( \sqrt{\frac{1}{64}} \).
This fractional exponent pattern is part of a broader rule, where \( a^{1/n} = \sqrt[n]{a} \). Mastering the translation between different forms (exponential to radical) widens your ability to manipulate and simplify expressions effectively.

Simplification

Simplification is the process of reducing an expression to its simplest form. This usually involves performing operations to make an expression easier to work with or understand.
In the context of our exercise, we simplify \( \sqrt{\frac{1}{64}} \) by taking the square root of both the numerator and the denominator separately:

• The square root of 1 is 1, because \( 1 \times 1 = 1 \).
• The square root of 64 is 8, since \( 8 \times 8 = 64 \).

Thus, the expression \( \sqrt{\frac{1}{64}} \) simplifies to \( \frac{1}{8} \).
Simplification often helps in solving equations more efficiently and understanding the properties of numbers and expressions more deeply. It's a skill that enhances problem-solving capabilities in mathematics.