/*! 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 30 Multiply and simplify. $$ \l... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 30

Multiply and simplify. $$ \left(-10 c^{2} d^{5}\right)\left(\frac{1}{2} c d^{7}\right) $$

Short Answer

Expert verified
-5c^3d^{12}

Step by step solution

01

Distribute the Constants

Multiply the constants in the expression. Here, the constants are \(-10\) and \(\frac{1}{2}\): \(-10 \times \frac{1}{2} = -5\).
02

Multiply the Variables with the Same Base

Multiply the variables with the same base by adding their exponents. For the \(c\) variable, add the exponents 2 and 1: \(c^2 \times c = c^{2+1} = c^3\). For the \(d\) variable, add the exponents 5 and 7: \(d^5 \times d^7 = d^{5+7} = d^{12}\).
03

Combine All Parts

Combine all the parts, the constant and the variables, to get the final simplified expression: \(-5 \times c^3 \times d^{12} = -5c^3d^{12}\).

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.

Polynomial Multiplication
Polynomial multiplication involves distributing each term in one polynomial to every term in another polynomial. In this exercise, we multiply \( -10 c^2 d^5 \) by \( \frac{1}{2} c d^7 \). Think of it as distributing each part and combining the results.
We follow specific steps to ensure we multiply both constants and variables correctly. Always remember to handle coefficients (numbers in front of variables) separately from the variables themselves. Keeping variables with the same base together helps simplify the process later on.
Exponent Rules
Exponent rules are critical when multiplying variables with exponents. Here, we apply the rule that states, when multiplying two powers with the same base, you add their exponents. For example, \( c^2 \times c \) becomes \( c^{2+1} = c^3 \). This rule simplifies variables effectively.
Ensure to only combine exponents of matching bases. In our problem, we deal with two sets of variables: \( c^2 \times c \) and \( d^5 \times d^7 \). By adding the exponents, we get \( c^3 \) and \( d^{12} \), respectively.
Algebraic Simplification
Algebraic simplification reduces expressions to their simplest form. It involves combining like terms and reducing coefficients. After distributing and addressing the exponents, combine all parts succinctly, like in the provided answer: \( -5c^3d^{12} \).
With polynomial expressions, simplification often includes these steps:
鈥� Multiply coefficients separately.
鈥� Apply exponent rules for variables.
鈥� Combine resultant terms seamlessly.
Practicing these steps regularly makes simplification more intuitive.
Constants Multiplication
Constants multiplication is straightforward: multiply the coefficients alone, ignoring variables initially. For \( -10 \times \frac{1}{2} \), the product is \( -5 \).
Always handle the constants multiplication first because they can affect the sign and scale of the final expression. Here, multiplying \( -10 \) (negative) with \( \frac{1}{2} \) (positive) results in a negative value \( -5 \).
This step is crucial before moving on to variables, ensuring clarity in the final expression.

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

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (5 \sqrt{z}-4)(3 \sqrt{z}+6) $$

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (5 \sqrt{2}-\sqrt{11})(5 \sqrt{2}+\sqrt{11}) $$

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (\sqrt{a+2}-\sqrt{a-2})^{2} $$

Perform the indicated operations and simplify. $$ (w+4)^{3} $$

Multiply and simplify. $$ 7 m^{2}\left(2 m^{4}-3 m+4\right) $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

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.