/*! 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 46 Multiply and, if possible, simpl... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 46

Multiply and, if possible, simplify. $$ \frac{5 t^{3}}{4 t-8} \cdot \frac{6 t-12}{10 t} $$

Short Answer

Expert verified
\( \frac{3 t^2}{4} \)

Step by step solution

01

Factor the Numerators and Denominators

First, factor all the numerators and denominators where possible. Notice that \(4t-8\) can be factored as \(4(t-2)\), and \(6t-12\) can be factored as \(6(t-2)\).
02

Rewrite the Expression

Rewrite the original problem with the factors: \[\frac{5 t^3}{4(t-2)} \times \frac{6(t-2)}{10t}\]
03

Cancel Common Factors

Identify and cancel out the common factors in the numerators and denominators. The term \(t-2\) cancels out: \[\frac{5 t^3}{4} \times \frac{6}{10t}\].
04

Simplify the Remaining Expression

Multiply the remaining expressions: \[\frac{5 t^3 \times 6}{4 \times 10t} = \frac{30 t^3}{40 t}\].
05

Reduce the Fraction

Simplify the fraction by dividing both the numerator and the denominator by 10: \[\frac{30 t^3}{40 t} = \frac{3 t^2}{4}\].

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.

Factoring Polynomials
Factoring polynomials makes complex expressions easier to manage. Start by identifying common factors in each term.
For example, in the expression \(4t-8\), both terms share a common factor of 4. This allows us to factor it as \(4(t-2)\).
Likewise, \(6t-12\) can be factored as \(6(t-2)\).
Common factors can often simplify an expression significantly.
Practice by finding the greatest common factor and splitting polynomials into their simplest forms. This way, you can manage algebraic expressions better and make subsequent operations like multiplication much easier.
Simplifying Expressions
Simplifying expressions involves combining like terms and canceling common factors.
In our exercise, after factoring polynomials, we rewrite the expression: \[ \frac{5 t^3}{4(t-2)} \times \frac{6(t-2)}{10t} \]\
Notice that \(t-2\) appears in both the numerator and denominator.
Cancel these common factors to get a simpler expression.
Now, we have \[ \frac{5 t^3}{4} \times \frac{6}{10t} \]\
Always look for terms common in both the numerator and denominator to cancel out. Simplifying expressions helps to make the equations less cluttered and easier to solve.
Reducing Fractions
Reducing fractions is the process of dividing out common factors from the numerator and the denominator.
In our final step, we need to reduce: \[ \frac{30 t^3}{40 t} \]\
Divide both terms by 10 to simplify: \[ \frac{30 t^3}{40 t} = \frac{3 t^2}{4} \]
Always check to see if there are more common factors to divide out.
Reducing fractions keeps expressions concise and easier to interpret.
Practice simplifying fractions to ensure you can spot common factors quickly and accurately. This skill is crucial in making complex problems more manageable.

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

Use a graphing calculator to check Example 3 by setting \(y_{1}=\left(2 x^{2}-7 x-15\right) /(x-5)\) and \(\left.y_{2}=2 x+3 . \text { Then use either ( table }\right)\) (after selecting the ZOOM ZINTEGER option) or ( table \()\) (with TblMin \(=0\) and \(\Delta \mathrm{Tbl}=1\) ) to show that \(y_{1} \neq y_{2}\) for \(x=5\).

Factor completely. $$ 7 y^{2}-28 \quad[5.5] $$

Simplify. $$ \frac{a^{3}-2 a^{2}+2 a-4}{a^{3}-2 a^{2}-3 a+6} $$

To check Example \(4,\) Kara graphs $$ y_{1}=\frac{7 x^{2}+21 x}{14 x} \text { and } y_{2}=\frac{x+3}{2} $$ since the graphs of \(y_{1}\) and \(y_{2}\) appear to be identical, Kara believes that the domains of the functions described by \(y_{1}\) and \(y_{2}\) are the same, \(\mathbb{R} .\) How could you convince Kara otherwise?

Find the variation constant and an equation of variation if y varies directly as \(x\) and the following conditions apply. \(y=2\) when \(x=5\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Mechanics 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.