/*! 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 75 Simplify. Do not use negative ex... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 75

Simplify. Do not use negative exponents in the answer. \(\frac{a^{-5} a^{-9}}{a^{-8}}\)

Short Answer

Expert verified
The simplified expression is \(\frac{1}{a^6}\).

Step by step solution

01

Apply the Rule for Multiplying Exponents

When multiplying powers with the same base, add the exponents. Thus:\[a^{-5} \cdot a^{-9} = a^{-5 + (-9)} = a^{-14}.\]
02

Set up the New Fraction

Replace the numerator of the original fraction with the result from Step 1:\[\frac{a^{-14}}{a^{-8}}.\]
03

Apply the Quotient Rule for Exponents

When dividing powers with the same base, subtract the exponents:\[\frac{a^{-14}}{a^{-8}} = a^{-14 - (-8)} = a^{-14 + 8} = a^{-6}.\]
04

Rewrite with Positive Exponents

Express the answer without negative exponents:\[a^{-6} = \frac{1}{a^6}.\]

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 mathematical concept. They express how many times a number, known as the base, is multiplied by itself. If you see a number written as \(a^n\), the base is \(a\) and the exponent is \(n\). This means you multiply \(a\) by itself \(n\) times. For example, \(2^3\) means \(2 \times 2 \times 2 = 8\). Exponents allow us to write large numbers succinctly and perform calculations more easily.
  • Base: The number that is being multiplied.
  • Exponent: The number which denotes how many times the base is used as a factor.
  • Example: \(3^4 = 3 \times 3 \times 3 \times 3 = 81\).
Understanding exponents is crucial because they appear throughout different areas of mathematics, including algebra, calculus, and beyond.
Negative Exponents
Negative exponents might seem daunting at first, but they are simple once you understand their meaning. A negative exponent represents the reciprocal of the base raised to the opposite positive exponent. In other words, \(a^{-n}\) means \(\frac{1}{a^n}\). Negative exponents invert the base and change the exponent to positive. This concept helps simplify expressions and solve equations.
Here鈥檚 a breakdown:
  • \(a^{-1} = \frac{1}{a}\)
  • \(a^{-3} = \frac{1}{a^3}\)
  • \(a^{-5} = \frac{1}{a^5}\)
Mastering negative exponents will broaden your understanding of more complex algebraic expressions.
Exponent Rules
Exponent rules make it easier to work with expressions involving powers. These rules include ways to multiply, divide, and power exponents.
  • Product Rule: When multiplying like bases, add the exponents: \(a^m \cdot a^n = a^{m+n}\).
  • Quotient Rule: When dividing like bases, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
  • Power Rule: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{m \cdot n}\).
  • Zero Exponent Rule: Any non-zero base raised to the power of zero equals one: \(a^0 = 1\), where \(a eq 0\).
By applying these exponent rules, you can simplify expressions effectively and avoid pitfalls when dealing with complex calculations.

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

Perform each division. $$ \frac{y^{3}+8}{y+2} $$

Perform each division. $$ \frac{30 y^{8}+40 y^{7}}{10 y^{6}} $$

When dividing \(x^{3}+1\) by \(x+1,\) why is it helpful to write \(x^{3}+1\) as \(x^{3}+0 x^{2}+0 x+1 ?\)

Perform the indicated operations. a. \(\left(2 b^{2} d\right)^{2}\) b. \(\left(2 b^{2}+d\right)^{2}\)

Perform each division $$ \frac{a^{8}+a^{6}-4 a^{4}+5 a^{2}-3}{a^{4}+2 a^{2}-3} $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Calculus

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.