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Chapter 1: Problem 106

Raising a Product or a Quotient to a Power $$ \left(\frac{3 x^{5}}{y^{-4}}\right)^{2} $$

Short Answer

Expert verified
The simplified expression is \ 9x^{10}y^{8}.

Step by step solution

01

Understand the Problem

The given expression is \(\left(\frac{3x^{5}}{y^{-4}}\right)^{2}\). We need to raise both the numerator and the denominator to the power of 2 and simplify the expression.
02

Apply the Power Rule to the Numerator

Raise the numerator \(3x^{5}\) to the power of 2: \((3x^{5})^{2}\). This results in \3^{2} \cdot \ (x^{5})^{2} = 9x^{10}.
03

Apply the Power Rule to the Denominator

Raise the denominator \(y^{-4}\) to the power of 2: \((y^{-4})^{2}\). This results in \ (y^{-4})^{2} = y^{-8}.
04

Simplify the Expression

Combine the results from Step 2 and Step 3: \[ \frac{9x^{10}}{y^{-8}} \]. Since \ y^{-8} \ can be written as \ \frac{1}{y^{8}} \, the simplified expression becomes: \ 9x^{10}y^{8}. \

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule
The power rule is a fundamental concept when dealing with exponents. It states that when you raise a power to another power, you multiply the exponents. For any base聽\(a\)聽and any exponents聽\(m\)聽and聽\(n\), this rule is written as: \[ (a^m)^n = a^{m \times n} \].
This rule helps simplify expressions by combining like terms. When applying the power rule to a product or a quotient, each factor in the product or quotient must be raised to the power separately.
For example: \[ (ab)^n = a^n \times b^n \] and \[ \frac{a}{b}^n = \frac{a^n}{b^n} \].
Understanding this concept is essential for simplifying expressions involving exponents.
Simplifying Expressions
Simplifying expressions involves reducing them into their simplest form. This process makes it easier to work with them in equations or further calculations.
For instance, in the given problem: \[ \text{Simplify} \left( \frac{3x^5}{y^{-4}} \right)^2 \].
Using the power rule, we raise both the numerator and the denominator to the power of 2:
  • \[ (3x^5)^2 = 3^2 \times (x^5)^2 = 9x^{10} \]
  • \[ (y^{-4})^2 = y^{-8} \]
Next, we'll simplify the expression: \[ \frac{9x^{10}}{y^{-8}} \].
Since \[ y^{-8} \] is equivalent to \[ \frac{1}{y^8} \], the expression becomes: \[ 9x^{10}y^{8} \].

By following these steps, we have simplified the expression. Simplifying helps us work with and understand complex mathematical problems more easily.
Exponents
Exponents are a way to represent repeated multiplication of the same number. For example, \[ x^5 \] means聽\(x\) multiplied by itself 5 times. Here are some important rules about exponents that you should know:
  • Product of Powers: \[ x^a \times x^b = x^{a+b} \]
  • Quotient of Powers: \[ \frac{x^a}{x^b} = x^{a-b} \]
  • Power of a Power: \[ (x^a)^b = x^{a \times b} \]
  • Negative Exponent: \[ x^{-a} = \frac{1}{x^a} \]
Applying these rules helps simplify expressions with exponents.
In our problem, we used the power of a power rule to raise \[ 3x^5 \] and \[ y^{-4} \] to the power of 2. Understanding these rules is crucial for working with exponents in mathematics.

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Most popular questions from this chapter

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ 43-(-9+2)^{2}+18 \div 6 \cdot(-2) $$

Write an equivalent expression using an associative law. $$ x+(2 y+5) $$

Calculate using the rules for order of operations. Use the commutative, associative, and distributive laws to show that \(5(a+b c)\) is equivalent to \(c(b \cdot 5)+a \cdot 5 .\) Use only one law in each step of your work.

To the student and the instructor: Writing exercises, denoted by \([\%,\) are meant to be answered using sentences. Because answers to many writing exercises will vary, solutions are not listed at the back of the book. Charlie insists that \(15-4+1 \div 2 \cdot 3\) is \(2 .\) What error is he making?

Translate to an algebraic expression. $$ \\{x | x=2 n, n \text { is an integer }\\} $$
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