/*! 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 154 Simplify each expression. $$ ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 154

Simplify each expression. $$ \left(2 p q^{4}\right)^{3}\left(5 p^{6} q\right)^{2} $$

Short Answer

Expert verified
200p^{15}q^{14}

Step by step solution

01

Apply the power to each factor in the first term

Distribute the exponent 3 to each factor inside the parentheses \(2pq^4\): \((2pq^4)^3 = (2^3)(p^3)(q^{4*3}) = 8p^3q^{12}\).
02

Apply the power to each factor in the second term

Distribute the exponent 2 to each factor inside the parentheses \(5p^6q\): \((5p^6q)^2 = (5^2)(p^{6*2})(q^2) = 25p^{12}q^2\).
03

Multiply the simplified terms

Multiply the results from Step 1 and Step 2: \((8p^3q^{12})(25p^{12}q^2)\). Combine the like terms: \(8 \cdot 25 = 200\), \(p^{3+12} = p^{15}\), \(q^{12+2} = q^{14}\). This gives us the final result: \[200p^{15}q^{14}\].

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
Simplifying algebraic expressions means making them as simple as possible. It often involves reducing the number of terms, combining like terms, and applying mathematical operations.
For instance, when given a complex expression with multiple terms and exponents, we aim to transform it into its simplest form.
This process often includes applying rules of exponents and combining like terms.
Given the expression \((2pq^4)^3(5p^6q)^2\), we need to simplify it by following systematic steps.
Power of a Product
The power of a product states that when you raise a product to an exponent, you raise each factor in the product to that exponent.
Mathematically, this is written as \((ab)^n = a^n b^n\). So, each factor inside the parentheses is raised to the exponent outside the parentheses.

**Example:**
In the expression \((2pq^4)^3\), each factor inside the parentheses is raised to the 3rd power:
  • \(2^3 = 8\)
  • \((p)^3 = p^3\)
  • \((q^4)^3 = q^{4*3} = q^{12}\)
Thus, \((2pq^4)^3 = 8p^3q^{12}\).

Similarly, for \((5p^6q)^2\):
  • \(5^2 = 25\)
  • \((p^6)^2 = p^{6*2} = p^{12}\)
  • \((q)^2 = q^2\)
Therefore, \((5p^6q)^2 = 25p^{12}q^2\).
Algebraic Manipulation
Algebraic manipulation refers to rearranging and simplifying algebraic expressions using various algebraic rules. It involves operations such as combining like terms, distributing, and factoring.
Steps like distributing exponents and combining like terms are key in simplifying complex expressions.

When we multiply expressions like \((8p^3q^{12})(25p^{12}q^2)\), we use rules of exponents and arithmetic:

**Combining coefficients:**
  • Multiply the numerical coefficients together: \(8\cdot 25 = 200\)
**Combining variables with exponents:**
  • Add exponents of like bases: \(p^{3+12} = p^{15}\), \(q^{12+2} = q^{14}\)
Thus, the expression \((8p^3q^{12})(25p^{12}q^2)\) simplifies to \[200p^{15}q^{14}\].
Understanding algebraic manipulation helps in making complex expressions manageable and finding their simplest form.

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

Square each binomial using the Binomial Squares Pattern. $$ \left(3 x^{2}+2\right)^{2} $$

Square each binomial using the Binomial Squares Pattern. $$ (q+12)^{2} $$

Jennifer thinks the quotient \(\frac{a^{24}}{a^{6}}\) simplifies to \(a^{4}\). What is wrong with her reasoning?

Find each product. $$ \left(r s-\frac{2}{7}\right)\left(r s+\frac{2}{7}\right) $$

Multiply. Use either method. $$ (3 q+1)\left(q^{2}-4 q-5\right) $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

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.