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Chapter 7: Problem 54

Simplify each of the following as completely as possible. $$\frac{(-a)^{4}}{-a^{4}}$$

Short Answer

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Step by step solution

01

Simplify the numerator

The numerator \[(-a)^{4}\] can be simplified. Since raising a negative number to an even power results in a positive number, \((-a)^{4} = a^{4}\).
02

Write the simplified fraction

Substitute the simplified numerator back into the fraction to get \(\frac{a^4}{-a^4}\).
03

Simplify the fraction

The fraction \(\frac{a^4}{-a^4}\) can be simplified because the terms in the numerator and the denominator are the same except for the sign. Simplifying, we get \(\frac{a^4}{a^4} \times \frac{1}{-1} = -1\).

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Key Concepts

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

powers of a number
Understanding powers of a number is essential in algebra. Powers, or exponents, indicate how many times a number is multiplied by itself. For example, \((a^2) = a \times a\). When dealing with negative bases, note that raising a negative number to an even power results in a positive number. Conversely, an odd power on a negative base will result in a negative number. Thus, \((-a)^4 = a^4\) because 4 is an even number.
simplifying fractions
Simplifying fractions involves making the numerator and denominator as small as possible while keeping the values equivalent. If you have \(\frac{a}{b}\), and both \a\ and \b\ share common factors, you can divide both the numerator and the denominator by the greatest common divisor. In the given exercise, the numerator and the denominator are both \a^4\ (except for the sign). Therefore, you can cancel \a^4\ from the top and bottom, simplifying the fraction.
negative exponents
Negative exponents indicate reciprocal operations. For example, \(a^{-n} = \frac{1}{a^n}\). When simplifying expressions with negative exponents, it's crucial to recognize this relationship. In the given exercise, negative exponents are used indirectly as the expression simplifies. However, understanding this concept helps in comprehending that flipping the base changes the sign in the exponent.
basic algebra
In basic algebra, fundamental operations include addition, subtraction, multiplication, division, powers, and roots. One common simplification technique is cancelling out like terms, as seen in \(\frac{a^4}{a^4}\). Understanding how to manipulate algebraic expressions is essential because it allows for solving more complex problems. In this exercise, recognizing that \a^4\ in both the numerator and denominator can cancel each other out simplifies the expression significantly, leading to the final value of \-1\.

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Most popular questions from this chapter

Do your computation using scientific notation. $$\frac{(480)(0.008)}{(0.24)(4000)}$$

A collection of 40 coins consists of dimes and quarters. If the total value of the coins is \(\$ 6.70,\) how many of each are there?

Estimate the answer without actually carrying out the computation and make the most appropriate choice. If you multiply \(2.08 \times 10^{-7}\) by \(5.14 \times 10^{6},\) the result is closest to (a) 10 (b) 100 (c) 1 (d) 0.1 (e) 0.01

Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$(3 x+7)(5 x+4)$$

Convert each number into standard notation. $$2.8 \times 10^{4}$$
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