/*! 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 3 Evaluate using the rules of expo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 3

Evaluate using the rules of exponents. $$\frac{(-4)^{8}}{(-4)^{5}}$$

Short Answer

Expert verified
Using the quotient rule for exponents, we can simplify the expression \(\frac{(-4)^{8}}{(-4)^{5}}\) as \((-4)^{8-5} = (-4)^3\). Evaluating the expression, we have \((-4) \times (-4) \times (-4) = -64\). So, the final answer is \(-64\).

Step by step solution

01

Identify the base and exponents in the given expression

In our expression, the base is \(-4\). The first exponent (numerator) is \(8\), and the second exponent (denominator) is \(5\).
02

Apply the quotient rule for exponents

Using the quotient rule, we can rewrite the expression as: \((-4)^{8-5}\), which is the same as \((-4)^3\).
03

Evaluate the expression

Now we just need to evaluate the expression \((-4)^3\). This can be rewritten as \((-4) \times (-4) \times (-4)\). Proceeding with the multiplication, we have: Positve multiplied with negative will be always negative. Aditionally, multiplying by the same negative number results in negative, positive and negative. \((-4) \times (-4) = 16\) \(16 \times (-4) = -64\) The final answer is:\[-64\]

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.

Base and exponents
Every expression that involves exponents consists of two parts: the base and the exponent. The base is the number that is being multiplied by itself, while the exponent tells us how many times the base is used as a factor.
In the given expression \( \frac{(-4)^{8}}{(-4)^{5}} \), \(-4\) is the base. This means that it is the number being repeated in multiplication.
  • The exponent in \((-4)^8\) is \(8\), telling us that \(-4\) is used eight times in multiplication.
  • In \((-4)^5\), the exponent is \(5\), indicating that \(-4\) is used five times in multiplication.
Understanding the role of base and exponents is crucial as it forms the foundation of applying any arithmetic operation like finding the power, multiplying, or simplifying expressions involving exponents.
Quotient rule for exponents
When you come across a division problem involving exponents with the same base, the quotient rule for exponents is extremely helpful. This rule states that when dividing like bases, you subtract the exponents.
The general formula is \( \frac{a^m}{a^n} = a^{m-n} \), where \(a\) is the base and \(m\) and \(n\) are the exponents.
In our case, applying the quotient rule to \( \frac{(-4)^{8}}{(-4)^{5}} \) gives us:
  • We subtract the exponents: \(8 - 5\).
  • Resulting in \((-4)^{3}\), meaning the base \(-4\) is now raised to the third power.
This simplification technique allows for easier computation by reducing the expression to a single exponent. Using this rule is a critical step in simplifying and evaluating complex expressions with exponents.
Evaluating expressions with exponents
Once an expression is simplified using the rules of exponents, you need to evaluate it to find its numerical value. Evaluating involves considering the base and the new exponent, and conducting the multiplication.
Let鈥檚 take \((-4)^{3}\), which means multiplying the base, \(-4\), by itself three times:
  • First, multiply \((-4) \times (-4)\) to get \(16\).
  • Then, multiply \(16 \times (-4)\) to arrive at \(-64\).
This step-by-step multiplication shows how the exponent instructs us to repeatedly use the base as a multiplier, leading to the final result. By following careful and logical procedures, you can evaluate any expression with exponents, ensuring accuracy and confidence in your math work.

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

Does \((y-3)^{3}=y^{3}-27 ?\) Why or why not?

Expand. $$\left(\frac{2}{3} n+4\right)^{2}$$

Divide. $$\frac{8 x^{4}+2 x^{3}-13 x^{2}-53 x+14}{2 x^{2}+5 x+7}$$

If \(P(t)=t^{3}-3 t^{2}+2 t+5,\) find a) \(P(4)\) b) \(P(0)\)

Find the following special products. \(-(6 a-b)(6 a+b)\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

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.