Problem 28 - Intermediate Algebra

Question

Evaluate each expression. See Example 1. $$ \left(\frac{1}{16}\right)^{1 / 2} $$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{1}{4} \).

Step 1: Understand the Expression

The expression \( \left( \frac{1}{16} \right)^{1/2} \) involves finding the square root of the fraction \( \frac{1}{16} \). This is because raising any number to the power of \( \frac{1}{2} \) is equivalent to finding its square root.

Step 2: Find the Square Root of the Denominator

The denominator of the fraction \( \frac{1}{16} \) is 16. The square root of 16 is 4, because \( 4 \times 4 = 16 \). Thus, the square root of the denominator is 4.

Step 3: Find the Square Root of the Numerator

The numerator of the fraction \( \frac{1}{16} \) is 1. The square root of 1 is 1, because \( 1 \times 1 = 1 \). Thus, the square root of the numerator is 1.

Step 4: Combine the Square Roots

Now that we have found the square roots of the denominator and numerator separately, we can write the square root of the entire fraction as \( \frac{1}{4} \) because the square root of \( \frac{1}{16} \) is \( \frac{1}{4} \).

Key Concepts

Fraction Operations

Fractions are expressions that represent a part of a whole. They consist of a numerator at the top and a denominator at the bottom. When dealing with fraction operations, it's crucial to understand the relationships between these parts and how to manipulate them.

• Adding and Subtracting Fractions: To add or subtract fractions, they must have the same denominator. Once fractions have a common denominator, you can add or subtract the numerators, keeping the denominator unchanged. For example, \(\frac{1}{4} + \frac{1}{4} = \frac{2}{4}\), which simplifies to \(\frac{1}{2}\).
• Multiplying Fractions: Multiply the numerators to get the new numerator and the denominators to get the new denominator. For instance, \(\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}\).
• Dividing Fractions: To divide by a fraction, multiply by its reciprocal. For example, \(\frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{4}{6}\), which simplifies to \(\frac{2}{3}\).

Understanding these operations allows you to effectively handle fractions in more complex mathematical expressions.

Exponents

Exponents are a compact way to express repeated multiplication of the same number. For example, \(3^2\) means \(3\times 3\), which equals 9.When you see an expression like \((\frac{1}{16})^{1/2}\), the exponent \(1/2\) indicates a square root. This is because raising a number to the power of \(1/2\) is the same as taking its square root.

• Product Rule: When multiplying two powers with the same base, add the exponents. E.g., \(x^a \times x^b = x^{a+b}\).
• Quotient Rule: When dividing two powers with the same base, subtract the exponents. For example, \(x^a / x^b = x^{a-b}\).
• Power Rule: To raise a power to another power, multiply the exponents: \((x^a)^b = x^{a \times b}\).

Exponents make it easier to work with large numbers and simplify expressions that involve these mathematical operations.

Simplifying Expressions

Simplifying expressions means rewriting them in a more concise form without changing their value. This process often involves combining like terms, reducing fractions, and applying the laws of exponents.Let's look at general steps involved in simplifying:

• Combine Like Terms: Look for terms that have the same variable and power and combine them. For example, in \(3x + 2x\), you can combine them to get \(5x\).
• Apply the Order of Operations: Follow the BIDMAS/BODMAS order — Brackets, Indices (Exponents), Division and Multiplication, Addition and Subtraction — to ensure the correct order of operations.
• Factorization: Factor out common factors in expressions to simplify them further.

While simplifying, recheck each step to ensure mathematical accuracy. With practice, these techniques become intuitive, offering a more precise and faster way to handle complex expressions.