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Chapter 0: Problem 4

Simplify the expression. $$ (\sqrt[5]{x})^{5} $$

Short Answer

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Step by step solution

01

- Understand the expression

The given expression is \(\left( \sqrt[5]{x} \right)^{5}\). This involves a fifth root and an exponent of 5.
02

- Recall the property of exponents and roots

We know that the property of exponents and roots states \( \left( \sqrt[n]{a} \right)^{n} = a \), where \(n \) is a positive integer. For this problem, \(n = 5 \).
03

- Apply the property to simplify

Using the property, we have \( \left( \sqrt[5]{x} \right)^{5} = x \).

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Key Concepts

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

Exponents
Exponents are an essential part of algebra. They show how many times a number, known as the base, is multiplied by itself. For example, in the expression \(a^n\), 'a' is the base, and 'n' is the exponent. The exponent tells you to take 'a' and multiply it by itself 'n' times. Exponents are used to simplify expressions and solve equations efficiently. Some core properties of exponents include:
  • Product of Powers: \(a^m \times a^n = a^{m+n}\)
  • Power of a Power: \((a^m)^n = a^{mn}\)
  • Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
Understanding these properties will help you manipulate and simplify expressions that involve exponents.
Roots
Roots are the opposite of exponents. They determine what number, when multiplied by itself a specific number of times, produces the given number. The most common root is the square root, but other roots like cube roots and fifth roots also exist. Roots can be written using radical symbols or exponents. For example:
  • Square Root: \(\sqrt{a} = a^{1/2}\)
  • Cube Root: \(\sqrt[3]{a} = a^{1/3}\)
  • Fifth Root: \(\sqrt[5]{a} = a^{1/5}\)
The expression from the exercise, \(\sqrt[5]{x}\), is asking for the number that, when raised to the power of five, equals 'x'. When simplifying, understanding how roots and exponents interact is crucial.
Algebraic Properties
Algebraic properties are rules that govern the operations involving algebraic expressions. These include properties of addition, multiplication, exponents, and roots. Some key properties include:
  • Distributive Property: \(a(b + c) = ab + ac\)
  • Associative Property: For addition, \((a + b) + c = a + (b + c)\), and for multiplication, \((a \times b) \times c = a \times (b \times c)\)
  • Commutative Property: For addition, \(a + b = b + a\), and for multiplication, \(a \times b = b \times a\)
In the exercise given, we used a specific property of exponents and roots: \(\left( \sqrt[n]{a} \right)^{n} = a\). This property simplifies expressions where the root and power cancel each other out, leaving just the base 'a' (or in our case, 'x'). Emphasizing these properties will make it easier to solve and simplify complex algebraic expressions.

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Most popular questions from this chapter

The total national expenditure for health care has been increasing since the year 2000 . For privately insured individuals in the United States, the following models give the total amount spent for health insurance premiums \(I\) (in \$ billions) and the total amount spent on other out-of-pocket health- related expenses \(P\) (in \$ billions). (Source: U.S. Centers for Medicare \& Medicaid Services, www.census.gov) \(I=45.58 x+460.1 \quad\) Total spent on health insurance premiums \(x\) years since 2000. \(P=10.86 x+191.5 \quad\) Other out-of-pocket health-related expenses \(x\) years since 2000. Determine the total expenditure for other health-related out-of-pocket expenses for the year \(2005 .\)

Multiply and simplify. $$ (2 x-5)(x+4) $$

Add or subtract as indicated and simplify. $$ \left(0.004 m n^{5}-0.001 m n^{4}+0.05 m n^{3}\right)-\left(0.003 m n^{5}+0.007 m n^{4}-0.07 m n^{3}\right) $$

Perform the indicated operations and simplify. $$ (w+4)^{3} $$

Multiply and simplify. $$ (w+7)(6 w-3) $$
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