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Chapter 7: Problem 109

Simplify. Assume that \(x \geq 0 .\) \(\sqrt[10]{x^{16}}\)

Short Answer

Expert verified
The simplified form is \(x^{1.6} or (x^{4/5})\)

Step by step solution

01

Understand the Radical Notation

Recognize that the expression \(\root(10)(x^{16})\) represents the tenth root of \(x^{16}\).
02

Convert to Exponential Form

Write the radical expression in exponential form. This gives \(x^{16/10}\).
03

Simplify the Fraction

Simplify the fraction \(16/10\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives \(x^{8/5}\).
04

Simplify Further

Further simplify by recognizing \(x^{8/5}\) can be written as \(x^{1.6}\), which simplifies to \((x^0.2)^8\) .
05

Final Simplified Form

The expression \(x^{1.6}\) simplifies directly. Because \(x \geq 0\), the final simplified form is \((x^{4/5})^2 = x^{1.6}\)

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Key Concepts

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

Radical Notation
Radical notation is a way to represent roots. For example, \(\root(10)(x^{16})\) means the tenth root of \({x^{16}}\). In radical notation, the number inside the root is called the radicand, and the number indicating the root (like 10 in this case) is called the index.
  • Radical notation forms: square root: \(\root(2)(x)\)
  • cube root: \(\root(3)(x)\)
  • general n-th root: \(\root(n)(x)\)
Understanding radical notation is crucial in algebra since it helps in expressing roots in a simple manner. This allows us to manipulate and simplify root expressions more effectively.
Exponential Form
Exponential form is a way to write expressions involving roots as powers or exponents. For example, the tenth root of \({x^{16}}\) can be written as \({x^{16/10}}\).
This conversion makes it easier to perform algebraic operations like multiplication and division. Some key points for converting to exponential form:
  • \(\root(n)(x^m) = x^{m/n}\)
  • For our exercise: \(\root(10)(x^{16}) = x^{16/10}\).
Exponential form allows us to apply properties of exponents to simplify expressions efficiently.
Simplifying Fractions
Simplifying fractions is about reducing them to their simplest form. When we have a fraction like \(\frac{16}{10}\), we can simplify it by dividing both the numerator and the denominator by their greatest common divisor (GCD). For \(\frac{16}{10}\), the GCD is 2.
  • Divide numerator and denominator by GCD: \(\frac{16}{10} = \frac{16 \/ 2}{10 \/ 2} = \frac{8}{5}\).

This makes the expression simpler and often easier to work with, as in \({x^{16/10} = x^{8/5}}\).
Properties of Exponents
The properties of exponents help in manipulating and simplifying expressions that involve powers. Key properties include:
  • Product of powers: \({a^m \cdot a^n = a^{m+n}}\)
  • Power of a power: \(({a^m}^n = a^{m \cdot n}}\)
  • Power of a product: \(({ab}^m = a^m \cdot b^m}\)

Using these properties lets us rewrite and further simplify expressions. For instance, \({x^{8/5}}\) can be expressed as a power of a power: \(({x^{1.6}} = (x^{0.2})^8}\). This breaks down the expression into simpler parts, making complex calculations more manageable.

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