/*! 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 47 Simplify. Write the answer with ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 47

Simplify. Write the answer with positive exponents only. $$\frac{c^{5}}{d^{-4}}$$

Short Answer

Expert verified
The simplified expression is \( c^5 d^4 \) with all positive exponents.

Step by step solution

01

Rewrite Negative Exponents

When simplifying expressions with negative exponents, start by rewriting the expressions using positive exponents. Recall that \(d^{-4} = \frac{1}{d^4}\). Thus, the given expression can be rewritten as: \[ \frac{c^5}{d^{-4}} = c^5 \times \frac{1}{d^{-4}} \] \( \frac{1}{d^{-4}} \) is equivalent to \(d^4\).
02

Apply the Rewritten Expression

Replace \( \frac{1}{d^{-4}} \) with \(d^4\) and multiply. \[ c^5 \times d^4 \]
03

Write Final Answer

The simplified expression with positive exponents is: \[ c^5 \times d^4 = c^5 d^4 \]

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.

Positive exponents
Positive exponents are straightforward and help us understand how many times to multiply a number by itself. For example, in the expression \( c^5 \), the base is \( c \), and the exponent is 5. This means that you multiply \( c \) by itself five times: \( c \times c \times c \times c \times c = c^5 \).
Positive exponents make it easy to express large numbers compactly. They help in computations and simplify algebraic expressions. Remember, any base raised to a positive exponent remains in its regular form.
Negative exponents
Negative exponents can be tricky, but they follow specific rules that make them manageable. A negative exponent indicates that the base should be on the other side of the fraction line. For instance, in \( d^{-4} \), the base is \( d \), and the exponent is -4. This can be rewritten as:
\
\[ \frac{1}{d^4} \].

Essentially, \( d^{-4} \) means you take the reciprocal of \( d^4 \). This helps turn complex expressions into more manageable forms. Always remember to convert negative exponents to positive ones by using their reciprocal.
Exponent rules
Exponent rules help us simplify expressions efficiently. Here are some key rules:
  • Product of Powers Rule: \( a^m \times a^n = a^{m+n} \). When multiplying bases that are the same, add the exponents.

  • Power of a Power Rule: \( (a^m)^n = a^{mn} \). When raising a power to another power, multiply the exponents.

  • Power of a Product Rule: \( (ab)^n = a^n b^n \). When an entire product is raised to a power, distribute the exponent to both bases.

  • Quotient of Powers Rule: \( \frac{a^m}{a^n} = a^{m-n} \). When dividing bases that are the same, subtract the exponents.

  • Negative Exponent Rule: \( a^{-n} = \frac{1}{a^n} \). Whenever you see a negative exponent, it means the reciprocal of the base raised to the positive exponent.
By mastering these rules, you can easily handle any expression involving exponents and simplify them correctly.

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

Evaluate the polynomial for the given values of the variable. \(3 c^{3}-c+8\) a. for \(c=-1\) b. for \(c=0.1\)

Simplify. $$k^{24} k^{36} k^{50}$$

Perform the indicated operations. $$\left(1.2 x^{2}+3.5 x-9.6\right)+\left(2.9 x^{2}-6.7 x-3.5\right)$$

Simplify the expression. $$(2 w)^{5}$$

Perform the indicated operations. $$\left(\frac{7}{8} a-4 b-\frac{1}{2} c\right)-\left(-\frac{1}{4} a+\frac{1}{2} b+\frac{3}{8} c\right)$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

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.