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Chapter 4: Problem 95

\(\frac{(-4 y)^{8}(-4 y)^{-8}}{(-4 y)^{-26}(-4 y)^{27}}\)

Short Answer

Expert verified
\( \frac{1}{-4y} \)

Step by step solution

01

- Apply the power rule for multiplication

Combine the powers of \((-4y)\) in the numerator: \((-4y)^{8}(-4y)^{-8} = (-4y)^{8 + (-8)} = (-4y)^{0} = 1\).
02

- Apply the power rule for multiplication in the denominator

Combine the powers of \((-4y)\) in the denominator: \((-4y)^{-26}(-4y)^{27} = (-4y)^{-26 + 27} = (-4y)^{1} = -4y\).
03

- Simplify the entire fraction

Divide the simplified values from the numerator and the denominator: \[ \frac{1}{-4y} \]

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Key Concepts

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

Power Rule
The power rule is a key concept in algebra used to simplify expressions involving exponents. This rule states that when multiplying two expressions that have the same base, you can add their exponents. In shorthand, this is expressed as \((a^m \times a^n = a^{m+n})\).

In our exercise, both the numerator and the denominator contain \((-4y)\) raised to different powers, which means we can apply the power rule. Here's how it works in detail:
  • Numerator: \((-4y)^{8}(-4y)^{-8}\) simplifies to \((-4y)^{8 + (-8)} = (-4y)^{0} = 1\).
  • Denominator: \((-4y)^{-26}(-4y)^{27}\) simplifies to \((-4y)^{-26 + 27} = (-4y)^{1} = -4y\).
This helps us transform complicated expressions into simpler forms that are much easier to work with.
Simplifying Expressions
Simplifying expressions involves making them as simple as possible. This often includes combining like terms, reducing fractions, and applying rules of exponents.

In our exercise, after applying the power rule, we manage to simplify the complex terms in both the numerator and the denominator. Let's break it down:
  • Numerator: The initial complex term \((-4y)^{8}(-4y)^{-8}\) is simplified to 1, because \((-4y)^0 = 1\).
  • Denominator: The term \((-4y)^{-26}(-4y)^{27}\) simplifies to \(-4y\), because \((-4y)^{1} = -4y\).
Both these steps are crucial in making the final expression easier to handle. Finally, we divide the simplified numerator by the simplified denominator to get \(\frac{1}{-4y}\). This is as simplified as the expression can get.
Fractions in Algebra
Working with fractions in algebra can be tricky, especially when they involve algebraic expressions. However, understanding the basics can make the process much more manageable.
  • Simplifying Fractions: In our exercise, we've simplified both the numerator and the denominator independently before dividing them. This makes the final fraction much simpler.
  • Negative Exponents: One common pitfall is dealing with negative exponents. Recall that a negative exponent means taking the reciprocal of the base. Hence, \((a^{-n} = \frac{1}{a^n})\). In our case, negative exponents were added following the power rule, simplifying the expression directly.
Therefore, our final expression is \(\frac{1}{-4y}\), showing that even complex fractions can become simple when broken down step by step.

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Most popular questions from this chapter

Graph each equation by completing the table of values. $$\begin{aligned} &y=x^{2}-4\\\&\begin{array}{c|c}\hline x & y \\\\\hline-2 & \\\\\hline-1 & \\\\\hline 0 & \\\\\hline 1 & \\\\\hline 2 &\end{array}\end{aligned}$$

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (3 r-2 s)^{4} $$

Perform each indicated operation. \(\left(-2 b^{6}+3 b^{4}-b^{2}\right)+\left(b^{6}+2 b^{4}+2 b^{2}\right)\)

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In 2017 , the world's fastest computer could perform 93,014,600,000,000,000 calculations per second. How many calculations could it perform per minute? Per hour? (Data from www.top500.org)
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