/*! 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 13 Evaluate each exponential. $$ ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 13

Evaluate each exponential. $$ 169^{1 / 2} $$

Short Answer

Expert verified
13

Step by step solution

01

Understand the Exponential Expression

The given expression is \(169^{1 / 2}\). This is an example of an exponential expression where the base is 169 and the exponent is \(\frac{1}{2}\).
02

Recognize the Meaning of the Exponent

The exponent \(\frac{1}{2}\) indicates that we need to find the square root of 169. Generally, \(a^{1/n}\) is equivalent to the \(n\)-th root of \(a\).
03

Find the Square Root

Calculate the square root of 169. We know that \(\sqrt{169} = 13\), because \(13 \times 13 = 169\).
04

Write the Final Answer

Thus, \(169^{1 / 2} = 13\).

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.

Exponents
Exponents are a fundamental concept in mathematics, representing how many times a number, known as the base, is multiplied by itself. An expression like \(b^n\) means that the number \(b\) (the base) is multiplied by itself \(n\) (the exponent) times.
Examples of exponents include:
  • \(2^3 = 2 \times 2 \times 2 = 8\)
  • \(5^2 = 5 \times 5 = 25\)
  • \(10^1 = 10\)
In the specific case of the provided exercise, the exponential expression is \(169^{1/2}\). Here, the base is 169, while the exponent is \(\frac{1}{2}\). Understanding exponents is crucial when dealing with more complex mathematical concepts like square roots, which are essentially exponents in disguise.
Square Roots
Square roots are another important concept in mathematics, closely related to exponents. The square root of a number \(a\) is a value that, when multiplied by itself, gives \(a\). This is often written as \(\sqrt{a}\).
For example:
  • \(\sqrt{25} = 5\), since \(5 \times 5 = 25\)
  • \(\sqrt{9} = 3\), since \(3 \times 3 = 9\)
In exponential notation, a square root can be expressed as \(a^{1/2}\). This is why in the exercise, \(169^{1/2}\) means \(\sqrt{169}\). Calculating this, we find that \(\sqrt{169} = 13\), because \(13 \times 13 = 169\).
So, whether we see an exponent of \(1/2\) or the square root symbol, they mean the same thing: we need to find the square root of the number.
Radical Expressions
Radical expressions involve roots, such as square roots, cube roots, etc. The radical symbol \(\sqrt{}\) is used to denote the root of a number. For example:
  • \(\sqrt{16} = 4\), because \(4 \times 4 = 16\)
  • \(\sqrt[3]{27} = 3\), because \(3 \times 3 \times 3 = 27\)
The number inside the radical symbol is called the radicand. A radical expression can sometimes be simplified by breaking down the radicand into its prime factors or recognizing it as a perfect square or cube.
This can make it easier to evaluate or simplify more complex expressions.
In the given exercise, \(169^{1/2}\) is rewritten as \(\sqrt{169}\), leading us to the simplified result of 13. Understanding radicals and how to work with them is essential for mastering further mathematical concepts.

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

Rationalize each denominator. Assume that all variables represent positive real numbers. $$ \frac{2 \sqrt{5 r}}{\sqrt{m^{3}}} $$

Simplify. Assume that all variables represent positive real numbers. \(\sqrt{256 z^{12}}\)

Simplify. Assume that all variables represent positive real numbers. \(-\sqrt{25 t^{6} s^{20}}\)

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{3}{4+\sqrt{5}} $$

Simplify. Assume that all variables represent positive real numbers. \(\sqrt{72 k^{2}}\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.