/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 In Exercises \(21-38\), rewrite ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 21

In Exercises \(21-38\), rewrite each expression with rational exponents. $$\sqrt{7}$$

Short Answer

Expert verified
The expression \(\sqrt{7}\) with rational exponents is \(7^{1/2}\).

Step by step solution

01

Identify the components of the square root

In this problem, the number under the square root sign, \(a\), is 7. Also, since it's a square root, the \(n\) is implicitly 2.
02

Apply the rule to convert to rational exponent

To rewrite the expression with a rational exponent, follow the rule \(\sqrt[n]{a^m} = a^{m/n}\). In this problem, since we only have 'a' and 'n', the power 'm' of \(a\) is implicitly 1. Thus, we can rewrite the expression as \(7^{1/2}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Square Roots
A square root is a special case of a root, usually denoted with the symbol \(\sqrt{}\). The main objective of finding a square root is to determine what number, when multiplied by itself, gives the original number. For example, the square root of 4 is 2, because \(2 \times 2 = 4\).
Square roots are commonly seen in two forms:
  • Simple Square Root: \(\sqrt{a}\) which means the square root of \(a\).
  • Perfect Square: A number like 9, whose square root is a whole number.
In simplifying expressions involving square roots, remember that the process can sometimes lead to rational exponents. For instance, rewriting \(\sqrt{7}\) as \(7^{1/2}\), combines the concept of square roots with exponential functions.
Exponent Rules
Exponent rules provide a framework for simplifying expressions involving powers and roots. They are essential for performing calculations in algebra. Here are a few fundamental exponent rules:
  • Product of Powers: \(a^m \times a^n = a^{m+n}\)
  • Power of a Power: \((a^m)^n = a^{m \cdot n}\)
  • Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
  • Zero Exponent: Any number raised to the power of zero is 1, \(a^0 = 1\) (provided that \(a\) is not zero).
  • Negative Exponent: \(a^{-m} = \frac{1}{a^m}\)
By using these rules, expressions with exponents, such as rational exponents, can be efficiently simplified. For example, the expression \(7^{1/2}\) was derived using exponent rules applied to square roots.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operators (like addition or multiplication). They are the building blocks of algebra.
In algebra, expressions can involve:
  • Constants: Numbers that do not change, such as 5 or 7.
  • Variables: Symbols that represent unknown values, such as \(x\) or \(y\).
  • Coefficients: Numbers multiplying the variables, like 3 in \(3x\).
  • Exponential Terms: Variables raised to a power, for instance, \(x^2\).
The ability to manipulate algebraic expressions is critical. When dealing with expressions involving roots, such as \(\sqrt{7}\), it's useful to rewrite them with exponents (like \(7^{1/2}\)) to simplify calculations and solve equations.
Rational Numbers
Rational numbers encompass any numbers that can be expressed as the fraction \(\frac{a}{b}\), where \(a\) is an integer and \(b\) is a non-zero integer. They include:
  • Whole Numbers: Examples are 0, 1, 2, 3, etc.
  • Fractions: Such as \(\frac{1}{2}\), \(\frac{3}{4}\), or \(-\frac{5}{6}\).
  • Negative Numbers: Numbers less than zero, e.g., -3, -7.
Rational numbers are an important part of rational exponents, like those used in the expression \(7^{1/2}\). Here, the exponent \(\frac{1}{2}\) is a rational number, illustrating how rational exponents can be expressed to simplify roots through algebraic manipulation.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let \(f(x)=x^{2} .\) Find \(f(\sqrt{a+1}-\sqrt{a-1})\)

In Exercises \(39-64,\) rationalize each denominator. $$\sqrt[3]{\frac{3}{x y^{2}}}$$

In Exercises \(75-92,\) rationalize each denominator. Simplify, if possible. $$\frac{8}{\sqrt{5}}$$

Use a graphing utility to solve each radical equation. Graph each side of the equation in the given viewing rectangle. The equation's solution set is given by the \(x\) -coordinate(s) of the point (s) of intersection. Check by substitution. $$\begin{aligned} &\sqrt{x}+3=5\\\ &[-1,6,1] \text { by }[-1,6,1] \end{aligned}$$

In Exercises \(105-112,\) add or subtract as indicated. Begin by rationalizing denominators for all terms in which denominators contain radicals. $$\sqrt[3]{25}-\frac{15}{\sqrt[3]{5}}$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.