/*! 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 106 Simplify the following. See Sect... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 106

Simplify the following. See Section \(5.1 .\) $$ \frac{27 y^{14}}{3 y^{7}} $$

Short Answer

Expert verified
The simplified expression is \(9y^{7}\).

Step by step solution

01

Divide Coefficients

Start by simplifying the coefficients. The coefficients are 27 in the numerator and 3 in the denominator. Divide 27 by 3 to get 9.
02

Apply the Quotient Rule for Exponents

For the variables, apply the quotient rule for exponents: If you have the same base, you can subtract the exponents. Since the base is y, the exponents are 14 in the numerator and 7 in the denominator. Subtract 7 from 14 to get 7. This gives us: \[ y^{14-7} = y^7 \]
03

Combine Results

After simplification of both the coefficients and the variables, combine the results: \[ \frac{27y^{14}}{3y^{7}} = 9y^{7} \]

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.

Simplifying Expressions
When you see a fraction involving algebraic terms, like \( \frac{27y^{14}}{3y^{7}} \), the goal is to make it as simple as possible. This is known as simplifying an expression. In simplifying, we aim to reduce the terms so that they are easier to work with, while maintaining the value of the expression.
  • Start by looking at both the numbers (coefficients) and the variables. Each part can often be simplified separately.
  • For the coefficient (the number part), you'll divide the numbers in the fraction. In this case, divide 27 by 3.
  • For the variable part, especially when exponents are involved, you'll often use rules of exponents to reduce them.
By following these steps, you transform a complex-looking fraction into something simpler, which in this example results in \( 9y^7 \).
Quotient Rule for Exponents
The quotient rule for exponents is a handy tool in algebra that helps simplify expressions involving powers of the same base. The rule states that when you divide like bases, you subtract the exponents.
  • For example, \( y^{14} \div y^{7} \) involves dividing powers of \( y \). According to the quotient rule, subtract the exponent in the denominator (7) from the exponent in the numerator (14).
  • This simplification results in \( y^{14-7} = y^7 \).
Using the quotient rule effectively reduces the complexity of expressions with exponents. Remember, it only applies when the bases are the same, as in this example where the base is \( y \).
Coefficients in Algebra
Coefficients are the numbers that multiply a variable, and they play a critical role in algebraic expressions. They're typically the first part of the expression you simplify in any algebraic fraction.
  • In our example \( \frac{27y^{14}}{3y^{7}} \), the coefficient of the numerator is 27, and the denominator is 3.
  • To simplify, you divide these coefficients as you would with normal numbers: \( 27 \div 3 = 9 \).
  • Once the coefficients are simplified, you can then focus on simplifying the algebraic variables using other rules, like the quotient rule for exponents.
Understanding how to manipulate coefficients is integral to mastering algebra, as they often precede the variables that need further simplification.

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

The perimeter of a square is \(\left(12 x^{3}+4 x-16\right)\) feet. Find the length of its side.

Multiply. $$(t+3)\left(t^{2}-5 t+5\right)$$

Multiply. See Section 5.1 $$ 3 x(2 x) $$

Express each as a product of polynomials in \(x\). Then multiply and simplify. Find the area of the rectangular canvas if its length is \((3 x-2)\) inches and its width is \((x-4)\) inches. IMAGE CANNOT COPY!

Simplify each expression. See Section \(5.1 .\) $$ \frac{x^{3} y^{6}}{x y^{2}} $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

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.