Problem 21 - Precalculus Mathematics for Calculus

Question

Evaluate each expression. (a) \(\sqrt{\frac{4}{9}}\) (b) \(\sqrt[4]{256}\) (c) \(\sqrt[6]{\frac{1}{64}}\)

Step-by-Step Solution

Answer

(a) \(\frac{2}{3}\), (b) 4, (c) \(\frac{1}{2}\).

Step 1: Simplify the expression for part (a)

To evaluate \( \sqrt{\frac{4}{9}} \), recognize that the square root of a fraction is the square root of the numerator over the square root of the denominator. Thus, the expression becomes \( \sqrt{4} \div \sqrt{9} \).

Step 2: Calculate the square roots for part (a)

Calculate \( \sqrt{4} = 2 \) and \( \sqrt{9} = 3 \). Hence, \( \sqrt{\frac{4}{9}} = \frac{2}{3} \).

Step 3: Simplify the expression for part (b)

To find \( \sqrt[4]{256} \), identify what number raised to the fourth power results in 256.

Step 4: Calculate the fourth root for part (b)

Recognize that \( 4^4 = 256 \). Therefore, \( \sqrt[4]{256} = 4 \).

Step 5: Simplify the expression for part (c)

To evaluate \( \sqrt[6]{\frac{1}{64}} \), express it as the sixth root of the fraction \( \frac{1}{64} \), meaning \( \left( \frac{1}{64} \right)^{1/6} \).

Step 6: Calculate the sixth root for part (c)

Calculate \( 2^6 = 64 \), thus \( \sqrt[6]{64} = 2 \). Therefore, \( \sqrt[6]{\frac{1}{64}} = \frac{1}{2} \).

Key Concepts

Square Roots

The concept of square roots is an essential part of understanding radical expressions. The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, \( \sqrt{4} = 2 \) because 2 times 2 equals 4. Square roots are often represented by the radical symbol \( \sqrt{} \), making it easily recognizable.

• When dealing with square roots of fractions, such as \( \sqrt{\frac{4}{9}} \), we find the square root of the numerator and the denominator separately. This results in \( \sqrt{4} \div \sqrt{9} = \frac{2}{3} \).
• Perfect squares are numbers whose square roots result in whole numbers. Examples include 1, 4, 9, 16, and so on, which correspond to \( \sqrt{1} = 1 \), \( \sqrt{4} = 2 \), \( \sqrt{9} = 3 \) and so forth.

Recognizing perfect squares helps in quickly simplifying square root problems since you can calculate them mentally without needing a calculator.

Fourth Roots

The fourth root of a number involves finding a value that, when raised to the power of 4, results in the original number. This can be denoted as \( \sqrt[4]{x} \). For example, the fourth root of 256 is 4 because when you multiply 4 by itself four times \( (4 \times 4 \times 4 \times 4) \) it equals 256.

• To calculate fourth roots easily, familiarity with perfect fourth powers is helpful. For instance, the numbers 16, 81, and 256 are perfect fourths of 2, 3, and 4 respectively.
• A quicker estimate can be performed by knowing \( 4^4 = 256 \), hence \( \sqrt[4]{256} = 4 \).

Finding the fourth root is essentially solving for a base number that is repeated as a factor four times, to equal the target number. Consistent practice with these values will turn identifying fourth roots into an effortless task.

Sixth Roots

The sixth root of a number finds a value which, when raised to the sixth power, equals the target number. Notation-wise, it's represented as \( \sqrt[6]{x} \). Understanding sixth roots can at first be demanding, but with some practice, it becomes straightforward.

• For instance, to solve \( \sqrt[6]{\frac{1}{64}} \), consider individual sixth roots of the numerator and denominator. Start by recognizing that \( 2^6 = 64 \), meaning \( \sqrt[6]{64} = 2 \).
• This concludes that \( \sqrt[6]{\frac{1}{64}} = \frac{1}{2} \), as the sixth root of 1 is 1 and of 64 is 2.

Working with sixth roots often requires breaking down a problem into factors, similar to other roots. Series of simplifications make the method reliable and accurate, ensuring these roots are handled with ease.