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Chapter 10: Problem 42

Find each root. Assume that all variables represent nonnegative real numbers. $$\sqrt{256 x^{8}}$$

Short Answer

Expert verified
The root is \( 16x^4 \).

Step by step solution

01

Understand the Problem

You are asked to find the roots of the expression \( \sqrt{256 x^8} \). This means simplifying the expression under the square root.
02

Simplify the Square Root

To simplify \( \sqrt{256 x^8} \), you need to take the square root of both 256 and \( x^8 \).
03

Calculate the Square Root of 256

The square root of 256 is 16, since \( 16^2 = 256 \).
04

Calculate the Square Root of \( x^8 \)

The square root of \( x^8 \) is \( x^{8/2} = x^4 \). This is because when you take the square root, you divide the exponent by 2.
05

Combine the Results

Combine the square roots calculated in Steps 3 and 4 to get \( \sqrt{256 x^8} = 16x^4 \).

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Key Concepts

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

Simplifying Radical Expressions
Simplifying radical expressions involves breaking down expressions under a square root to their simplest form. This process makes calculations easier and the expression clearer. To simplify radicals:
  • Identify factors or perfect powers under the root that can be simplified. For example, in \( \sqrt{256 x^8} \), the number 256 and the expression \( x^8 \) are both perfect powers.
  • Calculate their roots separately. Find the square root of 256, which is 16, because \( 16^2 = 256 \). Similarly, for variables like \( x^8 \), take the root by dividing the exponent by 2, giving us \( x^4 \).
  • Combine the results. Join all simplified parts back under the same expression: \( \sqrt{256 x^8} = 16x^4 \).
Breaking expressions down into smaller parts and handling each part separately is a key strategy in simplifying radical expressions.
Exponents
Exponents indicate how many times to multiply a number by itself. They simplify repeated multiplication, making large computations more manageable. For instance, in \( x^8 \), the exponent 8 tells you that \( x \) is multiplied by itself 8 times.
  • Division of exponents: When simplifying roots, divide the exponent by 2 to find the square root. For \( x^8 \), dividing 8 by 2 gives \( x^4 \).
  • Multiplication property: Multiplying like bases with exponents adds the exponents, such as \( x^a \times x^b = x^{a+b} \).
  • Power of a power: Raising a power to a power multiplies the exponents, like \( (x^a)^b = x^{ab} \).
Understanding these properties of exponents can help transform complex expressions into simple, cleaner forms.
Nonnegative Real Numbers
Real numbers include all rational and irrational numbers. When we focus on nonnegative real numbers, these are numbers that are greater than or equal to zero.
  • Nonnegativity: This means these numbers do not include negative values, but zero is included.
  • Significance in expressions: In radical expressions like \( \sqrt{256 x^8} \), assuming variables represent nonnegative real numbers ensures the results of roots and powers are defined real numbers.
  • Provides validity: By assuming nonnegative numbers, it removes complexities involved with negative square roots or imaginary numbers.
Using nonnegative real numbers adds a level of simplicity and precision in solving mathematical problems involving roots and exponents.

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Most popular questions from this chapter

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8},\) find the following function values. $$g(1)$$

Fill in each box with the correct expression $$ \frac{\square}{y^{-3 / 4}}=y^{4 / 4}, \text { or } y $$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[8]{x^{4} y^{4}} $$

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(3 x^{1 / 4}\right)^{3}}{x^{1 / 12}} $$

Fill in each box with the correct expression $$ \square \cdot a^{2 / 3}=a^{3 / 3}, \text { or } a $$
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