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Chapter 5: Problem 82

Simplify each expression. See Examples 1 through 11. $$ \left(\frac{p t}{3}\right)^{3} $$

Short Answer

Expert verified
The expression simplifies to \( \frac{p^3 t^3}{27} \).

Step by step solution

01

Understand the Problem

We are given the expression \( \left(\frac{p t}{3}\right)^{3} \) and our task is to simplify it. Simplifying means expressing it in a form that is easier to interpret or evaluate.
02

Apply the Power of a Quotient Rule

The Power of a Quotient Rule states that \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \). Apply this to the given expression: \( \left(\frac{p t}{3}\right)^3 = \frac{(p t)^3}{3^3} \).
03

Apply the Power of a Product Rule

The Power of a Product Rule states that \( (ab)^n = a^n b^n \). Apply this rule to \( (p t)^3 \): \( (p t)^3 = p^3 t^3 \).
04

Compute the Exponents

Now calculate the individual exponents: \( 3^3 = 27 \). Thus the expression becomes \( \frac{p^3 t^3}{27} \).
05

Final Simplification

Write the expression in its simplest form as \( \frac{p^3 t^3}{27} \). This is the simplified form of the original expression \( \left(\frac{p t}{3}\right)^{3} \).

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Key Concepts

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

Power of a Quotient Rule
When working with algebra, simplifying expressions is a crucial skill. One powerful tool we can use is the power of a quotient rule. This rule helps us simplify expressions that involve a fraction raised to a power. Here鈥檚 how it works: If you have an expression of the form \( \left( \frac{a}{b} \right)^n \), you can rewrite it as \( \frac{a^n}{b^n} \). This means you're raising both the numerator and the denominator to the power of \( n \).

For example, in the expression \( \left( \frac{p t}{3} \right)^3 \), we apply this rule as follows:
  • Identify the numerator: \( p t \).
  • Identify the denominator: \( 3 \).
  • Raise both to the power of 3: \( \frac{(p t)^3}{3^3} \).
This step simplifies the problem, setting us up to apply additional rules as needed to finish the simplification.
Power of a Product Rule
The power of a product rule is another handy rule when dealing with expressions that have multiple factors raised to a power. The rule says that for any expression \( (ab)^n \), you can expand it to \( a^n b^n \). This is particularly useful when the base consists of multiple terms, like products, that are grouped together.

In our example, \( (p t)^3 \) can be simplified by:
  • Taking the first factor \( p \) and raising it to the power: \( p^3 \).
  • Taking the second factor \( t \) and raising it to the power: \( t^3 \).
By expanding the expression this way, the terms are easier to handle, and each factor is independently raised to the power. As a result, \( (p t)^3 \) becomes \( p^3 t^3 \). This makes it much simpler to substitute back into our simplified fraction.
Exponents Calculation
Calculating exponents is a key part of simplifying expressions in algebra. It involves multiplying a base by itself a certain number of times. Understanding how to perform exponent calculations can make a big difference in solving algebra problems efficiently.

In our presented problem, after applying the previous rules, we need to compute the exponent of our constant term. Specifically:
  • Calculate \( 3^3 \), which equals \( 27 \).
With these calculated values, the expression \( \frac{p^3 t^3}{27} \) represents the fully simplified version of the original expression. Performing the exponentiation accurately ensures that our algebraic simplification is correct and complete.

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Most popular questions from this chapter

Multiply. See Section 5.1 $$ -z^{2} y(11 z y) $$

Multiply each expression. $$ 9 a b\left(a b^{2} c+4 b c-8\right) $$

Perform the indicated operations. $$(2 x-1)(10 x-7)$$

Perform each indicated operation. $$ \left[\left(1.2 x^{2}-3 x+9.1\right)-\left(7.8 x^{2}-3.1+8\right)\right]+(1.2 x-6) $$

In which of the following is \(\frac{a+7}{7}\) simplified correctly? See the Concept Check in this section. a. \(a+1\) b. \(a\) c. \(\frac{a}{7}+1\)
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