/*! 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 59 Simplify each expression. Write ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 59

Simplify each expression. Write answers using positive exponents. $$ m^{-4} \cdot m^{-6} $$

Short Answer

Expert verified
\( \frac{1}{m^{10}} \)

Step by step solution

01

Apply the Product Rule of Exponents

When multiplying two powers with the same base, you add the exponents. Here, we have the expression \( m^{-4} \cdot m^{-6} \). According to the product rule: \( m^a \cdot m^b = m^{a+b} \). Therefore: \( m^{-4} \cdot m^{-6} = m^{-4 + (-6)} = m^{-10} \).
02

Convert to Positive Exponents

If you have a negative exponent, you can convert it to a positive exponent by taking the reciprocal of the base. The expression \( m^{-10} \) can be rewritten as \( \frac{1}{m^{10}} \): \( m^{-10} = \frac{1}{m^{10}} \).

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.

Product Rule of Exponents
The Product Rule of Exponents is a fundamental rule which makes working with powers easy and efficient. When multiplying two expressions with the same base, you combine the exponents by adding them. This rule can be simply stated as:
  • When you have two or more terms like this: \( a^m \cdot a^n \),
  • The result is a single term of the form: \( a^{m+n} \).
This means you simplify the expression by adding the two exponents.
If you encounter the expression like \( m^{-4} \cdot m^{-6} \), apply the product rule by adding the exponents, just like adding numbers:
\( (-4) + (-6) = -10 \).
So, \( m^{-4} \cdot m^{-6} = m^{-10} \). By mastering this simple rule, handling more complex expressions becomes much easier.
Negative Exponents
Negative exponents can seem tricky, but they are just a way to represent dividing by that base multiple times. A negative exponent doesn't mean a negative number; rather, it turns the base into a fraction. The principle is:
  • \( a^{-n} = \frac{1}{a^n} \)
This means that if you have a negative exponent, you can simplify it by moving it to the denominator and changing the exponent to a positive.
For example, \( m^{-10} \) can be expressed as \( \frac{1}{m^{10}} \).
This conversion not only makes the expression easier to manage but aligns with standard mathematical practice to express all final answers with positive exponents.
Positive Exponents
Positive exponents represent straightforward multiplication. When you see \( a^n \), it simply means multiply the base \( a \) by itself \( n \) times. This results in a larger number if the base is greater than one and non-zero.
  • \( a^n = a \times a \times \ldots \times a \) (n times)
Using positive exponents makes expressions cleaner and helps avoid fractions when expressing final results.
Applying these concepts ensures mathematical expressions are presented neatly and in accordance with standardized arithmetic conventions.
Even if you start with negative exponents, converting to positive ensures a more intuitive understanding and presentation of results, as in simplifying \( m^{-10} \) to \( \frac{1}{m^{10}} \).

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

During a performance, a juggler tosses one ball straight upward while continuing to juggle three others. The height \(f(t),\) in feet, of the ball is given by the polynomial function \(f(t)=-16 t^{2}+22 t+4\) where \(t\) is the time in seconds since the ball was thrown. Find the height of the ball 1 second after it is tossed upward. (PICTURE NOT COPY)

Multiply. Assume \(n\) is a natural number. $$ \left(a^{3 n}-b^{3 n}\right)\left(a^{3 n}+b^{3 n}\right) $$

Explain why the terms \(5 x^{2} y\) and \(5 x y^{2}\) are not like terms.

Explain why \(5 x^{2}+3 x+2=2 x(x+1)+3 x^{2}\) is not a quadratic equation.

Factor the polynomial in part a. Then use your answer to part a to find the remaining factorizations. (No new work is necessary!). a. \(3 a^{2}+12 a+12\) b. \(3 a^{2}+12 a b+12 b^{2}\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

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.