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Chapter 10: Problem 6

Perform the indicated operations. $$p^{3} p^{9} p$$

Short Answer

Expert verified
The result is \(p^{13}\).

Step by step solution

01

Understand Exponent Rules

When multiplying expressions with the same base, add the exponents. The formula to use is: \[a^m \times a^n = a^{m+n}\]
02

Identify the Base and Exponents

In this expression, the base is \(p\). The exponents that need to be added are 3, 9, and 1 (since \(p\) is equivalent to \(p^1\)).
03

Add the Exponents

Add the exponents together: \[3 + 9 + 1 = 13\]
04

Rewrite the Expression

Combine the base and the new exponent: \[p^{3+9+1} = p^{13}\]

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Key Concepts

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

multiplication of exponents
When you multiply expressions that have the same base, you add the exponents. This is one of the most important rules in working with exponents. The rule can be written as \( a^m \times a^n = a^{m+n} \). This means that if you have two numbers with the same base, you can simplify the expression by adding their exponents together.
For example: \[ x^2 \times x^3 = x^{2+3} = x^5 \]
This rule works because multiplying exponents means combining their effects, so it's helpful to combine them into a single exponent.
base and exponents
In any expression involving exponents, you have a base and an exponent. The base is the number that is being repeatedly multiplied. The exponent tells you how many times the base is being multiplied by itself.
For example, in the expression \( p^3 \), \( p \) is the base, and 3 is the exponent. This means \( p \) is multiplied by itself 3 times: \[ p \times p \times p \]
Recognizing the base and the exponents is crucial when you are simplifying expressions involving exponents.
combining exponents
Combining exponents involves adding them when they share the same base. Let's walk through an example. If you have an expression like \( p^3 \times p^9 \times p \), you should:
1. Identify the base: Here, the base is \( p \).
2. Identify the exponents: In this case, the exponents are 3, 9, and 1 (since \( p \) is the same as \( p^1 \)).
3. Add the exponents together: \[ 3 + 9 + 1 = 13 \]
This gives you the simplified form of the expression: \ p^{13} \
By combining the exponents, you turn a complicated multiplication into a simple expression with one exponent.

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Most popular questions from this chapter

Subtract \(\left(9 x^{2}+16 x-4\right)\) from \(\left(2 x^{2}-4 x-9\right)\)

Simplify the expression. $$\left(\frac{7 c}{d^{3}}\right)^{2}$$

Simplify. Write the answers with positive exponents only. $$\left(x y^{-4}\right)^{3}$$

The cost (in dollars) for a cab ride is given by the polynomial \(2+1.25 x+\) \(0.25 y .\) In this context, \(x\) is the number of miles driven and \(y\) is the time in minutes for the ride. a. Evaluate the polynomial for \(x=10\) and \(y=20 .\) Interpret the answer in the context of this problem. b. Evaluate the polynomial for \(x=22\) and \(y=40 .\) Interpret the answer in the context of this problem.

Simplify the expression. $$(4 x y)^{3}$$
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