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Chapter 0: Problem 11

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

Short Answer

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

Step by step solution

01

Evaluate the Square Root

To evaluate \(\sqrt{\frac{4}{9}}\), we first find the square roots of the numerator and the denominator separately. The square root of 4 is 2, and the square root of 9 is 3. Therefore, \(\sqrt{\frac{4}{9}} = \frac{2}{3}\).
02

Evaluate the Fourth Root

To find \(\sqrt[4]{256}\), we need to determine what number raised to the power of 4 equals 256. \(256 = 2^8\), and \((2^2)^4 = 2^8\). Thus, \(\sqrt[4]{256} = 2^2 = 4\).
03

Evaluate the Sixth Root

To evaluate \(\sqrt[6]{\frac{1}{64}}\), we find the sixth root of the numerator and the denominator separately. Since \(1\) to any power is still 1, \(\sqrt[6]{1} = 1\). For 64, note that \(64 = 2^6\) and \(\sqrt[6]{64} = \sqrt[6]{2^6} = 2\). Hence, \(\sqrt[6]{\frac{1}{64}} = \frac{1}{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Roots
A **square root** of a number is a value that, when multiplied by itself, gives you the original number. It's denoted using the radical symbol \( \sqrt{} \). For example, \( \sqrt{9} = 3 \) because when you multiply 3 by itself \( (3 \times 3) \), you get 9.
Square roots are a common operation in algebra, and understanding them is key to solving more complex equations.
When you need to find the square root of a fraction, such as \( \sqrt{\frac{4}{9}} \), you take the square root of the numerator and the square root of the denominator separately:
  • Square root of 4 is 2.
  • Square root of 9 is 3.
Thus, \( \sqrt{\frac{4}{9}} = \frac{2}{3} \). This approach makes dealing with fractions much simpler when evaluating square roots.
Fourth Roots
**Fourth roots** involve finding a number that, when raised to the power of four, equals the original number. It鈥檚 notated as \( \sqrt[4]{} \). Calculating a fourth root can seem challenging, but breaking it down into steps can simplify the process.
Consider \( \sqrt[4]{256} \). Here, we look for a number that raised to the power 4 equals 256.
We start with the observation:
  • \( 256 = 2^8 \).
  • Now, this can also be expressed as \( (2^2)^4 = 2^8 \), indicating that two squared equals four and four multiplies itself fully to result in two to the eighth power.
Hence, \( \sqrt[4]{256} = 2^2 = 4 \). Fourth roots often seem tricky, but understanding exponent rules makes them more approachable.
Sixth Roots
**Sixth roots** represent a number that, when used as a factor six times, results in the original number. This operation is expressed with \( \sqrt[6]{} \) notation and requires patience and some fundamental understanding of exponents.
To compute something like \( \sqrt[6]{\frac{1}{64}} \), break it down:
  • Finding the sixth root of 1 is simple, as 1 raised to any power remains 1, so \( \sqrt[6]{1} = 1 \).
  • Addressing 64: \( 64 = 2^6 \).
  • Then, the sixth root of \( 64 \) is \( \sqrt[6]{64} = \sqrt[6]{2^6} = 2 \).
Therefore, \( \sqrt[6]{\frac{1}{64}} = \frac{1}{2} \). Sixth roots might seem daunting at first, but by applying exponent rules, they become much easier to manage.

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