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Chapter 10: Problem 14

Use radical notation to write each expression. Simplify if possible. $$ -64^{1 / 2} $$

Short Answer

Expert verified
The simplified expression is \(-8\).

Step by step solution

01

Identify the Expression

The expression given is \(-64^{1/2}\). This is the notation for a power or exponent where \(64\) is raised to the power of \(1/2\).
02

Understand Radical Notation

Any expression in the form of \(a^{1/n}\) can be written as \(\sqrt[n]{a}\). Hence, \(64^{1/2}\) can be rewritten using radical notation as \(\sqrt{64}\).
03

Simplify the Square Root

The square root \(\sqrt{64}\) means what number multiplied by itself equals \(64\). We know that \(8^2 = 64\). Therefore, the square root of \(64\) is \(8\).
04

Apply the Negative Sign

Remember that the original expression was \(-64^{1/2}\). We simplified \(64^{1/2}\) to \(8\) in the previous step. Therefore, by applying the negative sign, the final simplified expression is \(-8\).

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Key Concepts

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

Simplifying Radicals
Simplifying radicals is the process of finding a simpler form of a radical expression. A radical expression often includes a number under a square root or another type of root. To simplify it, you need to find the highest perfect square (or cube, etc.) factor of the number under the root and simplify accordingly.

Here鈥檚 a simple guide:
  • Identify if a number inside the radical can be broken down into parts.
  • Look for perfect square factors.
  • Represent those parts as a product and simplify the radicals.
  • Combine like terms if more than one radical remains.
By following these steps, you can break down complex radical expressions into their simplest form, making them easier to work with in further calculations.
Square Roots
A square root is a special mathematical operation. It finds the number that, when multiplied by itself, gives the original number. For example, the square root of 64, denoted as \( \sqrt{64} \), is 8 because \(8 \times 8 = 64\).

To understand square roots, consider:
  • Every positive number has a positive and a negative square root. However, typically, we refer to the positive one as the principal square root.
  • Some numbers, like 4 and 9, are termed "perfect squares" because their square roots are whole numbers (2 and 3 respectively).
  • If the number is not a perfect square, its square root can still be approximated or left as a radical.
Understanding square roots is crucial because they pop up frequently in various math problems and scenarios.
Exponents as Fractions
Exponents describe how many times to multiply a number by itself. When the exponent is a fraction, it represents a root. For example, an exponent of \(\frac{1}{2}\) indicates a square root, like in \(64^{1/2}\).

Key points about fractional exponents include:
  • The numerator of the fraction is the power that the base number is raised to, while the denominator indicates what root to take.
  • A fractional exponent such as \(a^{1/n}\) can be rewritten as \(\sqrt[n]{a}\).
  • This provides a straightforward way to convert between radical form and exponential form, which can be particularly handy for solving equations.
Understanding how to work with fractional exponents allows you to maneuver between different forms of mathematical expressions seamlessly.

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Most popular questions from this chapter

Use rational expressions to write as a single radical expression. $$ \sqrt[3]{b} \cdot \sqrt[5]{4 a} $$

Basal metabolic rate \((B M R)\) is the number of calories per day a person needs to maintain life. A person's basal metabolic rate \(B(w)\) in calories per day can be estimated with the function \(B(w)=70 w^{1 / 4},\) where \(w\) is the person's weight in kilograms. Use this information to answer Exercises. Estimate the BMR for a person who weighs 60 kilograms. Round to the nearest calorie. (Note: 60 kilograms is approximately 132 pounds.)

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