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Chapter 7: Problem 63

Simplify. \(-\sqrt[4]{512}\)

Short Answer

Expert verified
-2^2.25

Step by step solution

01

- Express the number under the radical

To simplify \(-\sqrt[4]{512}\), first express 512 as a power of a smaller base. We know that 512 can be written as 512 = 2^9.
02

- Apply the fourth root

When taking the fourth root of \(2^9\), we get \((-\sqrt[4]{2^9})\). Recall that \(\sqrt[n]{a^m} = a^{m/n}\), so this becomes \(-2^{9/4}.\)
03

- Simplify the exponent

Simplify the exponent \(-2^{9/4}\). This term cannot be simplified further unless a decimal approximation is given. Hence, the simplified form of the expression is \(-2^{2.25}.\)

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Key Concepts

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

Radical Expressions
A radical expression includes a radical sign (鈭�) and a number or expression inside it, called the radicand. Radicals simplify the expressions involving roots, such as square roots and cube roots. To work with radical expressions, you need to understand the rules of radicals:
  • When you multiply two radicals, you can multiply the radicands: \sqrt{a} \times \sqrt{b} = \sqrt{a \times b}.
  • When you divide two radicals, you can divide the radicands: \frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}.
These rules help simplify expressions with radicals. For instance, to simplify \(-\sqrt[4]{512}\), you need to express 512 as a power of a smaller base (\(2^9\)). This makes it easier to apply the root.
Exponents
Exponents are shorthand for repeated multiplication of the same number. For example, 2^3 means 2 \times 2 \times 2. Here are some important properties of exponents:
  • When you multiply like bases, you add the exponents: a^m \times a^n = a^{m+n}.
  • When you divide like bases, you subtract the exponents: \frac{a^m}{a^n} = a^{m-n}.
These properties are useful when simplifying expressions with exponents. In our original exercise, we simplified 2^9 and took its fourth root (\sqrt[4]{2^9}). Applying the rule \(\sqrt[n]{a^m} = a^{m/n}\), we get {2^{9/4}}.
Roots
Roots are the inverse operation of powers. If you know how to work with exponents, it will help you understand roots. For example, the square root of a number is the value that, when squared, gives the original number. Here are some key points about roots:
  • The nth root of a number can be written as \sqrt[n]{a}. For example, \(\sqrt[3]{8} = 2\).
  • Roots can often be simplified by expressing the radicand using exponents, such as converting 512 to 2^9 before applying the fourth root.
In our problem, we used the fourth root. Applying the property of roots and exponents, we have \sqrt[4]{2^9} = 2^{9/4}. This makes dealing with radicals easier.

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