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Chapter 1: Problem 22

Divide and simplify. $$ \frac{a^{11}}{a^{8}} $$

Short Answer

Expert verified
The simplified form is \(a^{3}\).

Step by step solution

01

Identify the Base and Exponents

In the fraction \(\frac{a^{11}}{a^{8}}\), the base is \(a\) and the exponents are 11 in the numerator and 8 in the denominator.
02

Apply the Division Rule for Exponents

Use the division rule for exponents, which states that when you divide like bases you subtract the exponent in the denominator from the exponent in the numerator: \(\frac{a^{11}}{a^{8}} = a^{11-8}\).
03

Simplify the Exponent

Subtract the exponent in the denominator from the exponent in the numerator: \(11 - 8 = 3\). This gives \(a^{3}\).
04

Write Final Answer

The simplified form of \(\frac{a^{11}}{a^{8}}\) is \(a^{3}\).

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Key Concepts

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

Base and Exponent Identification
When working with exponents, it is crucial to identify the base and the exponent correctly. The base is the number or variable that is being multiplied by itself, and the exponent tells us how many times to multiply the base. For example, in the expression \(a^{11}\), \(a\) is the base, and 11 is the exponent. Understanding this will help make the manipulation of expressions involving exponents much more straightforward.
Division Rule for Exponents
The division rule for exponents is one of the fundamental rules in algebra. It states that when you divide two expressions with the same base, you subtract the exponent in the denominator from the exponent in the numerator. For instance, in the expression \(\frac{a^{11}}{a^{8}}\), both the numerator and the denominator have the same base \(a\). According to the division rule, you subtract the exponent in the denominator (8) from the exponent in the numerator (11) to get the simplified form: \(\frac{a^{11}}{a^{8}} = a^{11-8}\).
Simplifying Exponents
Simplifying exponents involves reducing complex expressions to their simplest form. After using the division rule for exponents, you'll often end up with a much simpler expression. Continuing from the division step, we get \(a^{11-8} = a^{3}\). Here, we have subtracted 8 from 11, leaving us with a new exponent of 3. Hence, \(\frac{a^{11}}{a^{8}}\) simplifies to \(a^{3}\).
Intermediate Algebra
In intermediate algebra, understanding how to manipulate exponents is essential. These skills are foundational for solving more complex algebraic expressions. To master these concepts, practice identifying bases and exponents, using division rules, and simplifying expressions regularly. This exercise, for example, not only involves basic arithmetic skills, like subtraction, but also the application of key algebraic rules. It's a step toward more advanced topics in algebra and beyond.

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Most popular questions from this chapter

Use roster notation to write each set. The set of all natural numbers that are multiples of 5

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ -(-2)^{2} $$

Insert one pair of parentheses to convert each of the following false statements into a true statement. $$ 2-7 \cdot 2^{2}+9=-11 $$

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.

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