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Chapter 7: Problem 65

Simplify. Use absolute-value notation when necessary. $$ \sqrt[9]{t^{9}} $$

Short Answer

Expert verified
The simplified form is \( |t| \).

Step by step solution

01

Identify the Expression

The expression given to simplify is \ \ \( \sqrt[9]{t^{9}} \).
02

Recall the Property of Radicals and Exponents

Recall the property that \ \( \sqrt[n]{a^{n}} = |a| \) when \ \( n \) is an odd integer. This indicates that any radical with an odd root and its corresponding exponent simplifies to the base with absolute value.
03

Apply the Property

Apply the property to the given expression: \ \( \sqrt[9]{t^{9}} = |t| \).

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Key Concepts

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

Understanding Radical Expressions
A radical expression is an expression that includes a root symbol. The radical symbol (鈭�) represents the square root, but there are other roots as well, such as cube roots (鲁鈭�) and fourth roots (鈦粹垰). In our exercise, we are dealing with the ninth root, shown as \( \sqrt[9]{...} \).

To simplify a radical expression, you often need to rewrite it in a form that is easier to understand. This process is known as simplifying the radical.

Simplifying radicals can involve rationalizing the denominator, combining like terms, or using properties of exponents and absolute values, which we will explore in the next sections.
Exponents and Their Properties
Exponents are a way to represent repeated multiplication of the same number. For instance, \( a^2 \) means \(a \) multiplied by itself, \( a \times a \).

When simplifying expressions involving exponents and radicals, it's essential to understand some key properties:
  • The power rule: \( (a^m)^n = a^{m \cdot n} \).
  • The product rule: \( a^m \cdot a^n = a^{m+n} \).
  • The quotient rule: \( \frac{a^m}{a^n} = a^{m-n} \).
Our exercise uses the rule that \[ \sqrt[n]{a^n} = |a| \] when \ n \ is an odd integer. This means taking the nth root of a number raised to the nth power returns the original number, but sometimes we need to use absolute value to ensure the result is positive.
The Concept of Absolute Value
The absolute value of a number is its distance from zero on the number line. It is always non-negative. The absolute value of \(a\), represented as \ |a| \, is \ a \ if \ a \ is positive or zero, and \ -a \ if \ a \ is negative.
In mathematical terms:
  • \ |a| = a \ if \ a \geq 0 \
  • \ |a| = -a \ if \ a < 0 \
Absolute value is crucial when dealing with radicals that have odd roots, such as in our exercise. It ensures the simplified expression accurately represents the original, even if the base is negative. Without absolute value, the simplified form might incorrectly suggest a different value or sign.

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Most popular questions from this chapter

Perform the indicated operations. $$ \frac{(\sqrt[3]{81 m n^{2}})^{2}}{(\sqrt[3]{m n})^{2}} $$

Simplify. $$ \frac{1}{4+\sqrt{3}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{3}-4} $$

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Solve. $$ \sqrt{2 x}-1=2 $$

Choose from the following list the word that best completes each statement. Not every word will be used. even isolate odd powers radical raise rational square roots A check is essential when we raise both sides of an equation to a(n) \(\longrightarrow\) power.
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