/*! 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 53 Evaluate each expression. $$ ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 53

Evaluate each expression. $$ \frac{1}{4^{-2}} $$

Short Answer

Expert verified
The value of the expression is 16.

Step by step solution

01

- Understand negative exponents

A negative exponent means that the base is on the wrong side of the fraction line, so you must flip it to make the exponent positive. For any nonzero number 'a' and integer 'n', we have \[a^{-n} = \frac{1}{a^n}\].
02

- Apply the property of negative exponents

Using the property of negative exponents, rewrite \[4^{-2}\] as \[4^{-2} = \frac{1}{4^2}\].
03

- Rewrite the original expression

Now replace \[4^{-2}\] in the original expression: \[\frac{1}{4^{-2}} = \frac{1}{\frac{1}{4^2}}\].
04

- Simplify the fraction

To simplify \[\frac{1}{\frac{1}{4^2}}\], multiply the numerator (1) by the reciprocal of the denominator (\[4^2\]): \[\frac{1}{\frac{1}{4^2}} = 4^2\].
05

- Solve the exponent

Calculate \[4^2\]: \[4^2 = 16\].

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.

Simplifying Fractions
When simplifying fractions, our goal is often to make them as straightforward as possible. This involves reducing fractions to their simplest form or making complex expressions easier to handle. In the original exercise, we had the fraction \(\frac{1}{4^{-2}}\). By understanding how negative exponents work, we can simplify this fraction efficiently.

Here鈥檚 a handy tip: When you see a fraction with a negative exponent in the denominator, you can simplify it by flipping it around, converting it from negative to positive. This operation leverages the properties of fractions and negative exponents, making your calculations easier and more intuitive.
Reciprocal
The concept of a reciprocal is vital when dealing with fractions and exponents. The reciprocal of a number is simply 1 divided by that number. Mathematically, if you have a number \'x\, the reciprocal is \(\frac{1}{x}\).

In the exercise solution, we encountered the expression \(\frac{1}{\frac{1}{4^2}}\). To simplify this, we used the reciprocal of \(\frac{1}{4^2}\), which is \(4^2\). So, \(\frac{1}{\frac{1}{4^2}} = 4^2\). Understanding reciprocals helps in easily converting complex fractions into manageable forms.
Evaluating Exponents
Evaluating exponents involves calculating the value of a base raised to a power. An exponent tells you how many times to multiply the base by itself.

In the given problem, we eventually needed to evaluate \(4^2\). Here, 4 is the base, and 2 is the exponent. This means multiplying 4 by itself: \(4 \times 4 = 16\).

Mastering exponent evaluation is crucial for tackling algebra problems efficiently. Remember: Practicing with different bases and exponents will bolster your understanding and make complex problems more approachable.

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

In Exercises 65鈥�74, the factors involve fractions or decimals. Apply the methods of this sec- tion, and find each product. $$ \left(k-\frac{5}{7} p\right)^{2} $$

Solve each problem. The U.S. budget first passed \(\$ 1,000,000,000\) in \(1917 .\) Seventy years later, in \(1987,\) it exceeded \(\$ 1,000,000,000,000\) for the first time. The budget request for fiscal-year 2009 was \(\$ 3,100,000,000,000 .\) If stacked in dollar bills, this amount would stretch \(210,385 \mathrm{mi},\) almost \(90 \%\) of the distance to the moon. Write the four boldfaced numbers in scientific notation.

Find each product. $$ (4 z-x)\left(z^{3}-4 z^{2} x+2 z x^{2}-x^{3}\right) $$

Evaluate. Express answers in standard notation. $$ \frac{20,000 \times 0.018}{300 \times 0.0004} $$

In Exercises 65鈥�74, the factors involve fractions or decimals. Apply the methods of this sec- tion, and find each product. $$ \left(4 x-\frac{2}{3}\right)\left(4 x+\frac{2}{3}\right) $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

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.