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Chapter 4: Problem 28

Find each product or quotient. Express using exponents. $$\frac{a^{8}}{a^{8}}$$

Short Answer

Expert verified
The result is 1.

Step by step solution

01

Recognize the Expression Forms

The expression given is a fraction with the same base in both the numerator and the denominator: \( \frac{a^{8}}{a^{8}} \). This is an example of a quotient of powers.
02

Apply the Law of Exponents

When dividing powers with the same base, subtract the exponents: \( a^m = a^{m-n} \). Therefore, \( \frac{a^{8}}{a^{8}} = a^{8-8} \).
03

Simplify the Subtraction

Subtract the exponents: \( 8 - 8 = 0 \). Substituting back, the expression becomes \( a^0 \).
04

Know the Rule for Zero Exponent

Any non-zero number raised to the power of zero equals 1. Therefore, \( a^0 = 1 \).

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Key Concepts

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

Quotient of Powers
In mathematics, when we encounter powers or exponents, it's essential to know how to manage basic operations between them. The quotient of powers rule is one such method, and it's pretty straightforward. When you divide two numbers with the same base and different exponents, you simply subtract the exponent in the denominator from the exponent in the numerator. For example:
  • Given: \( \frac{a^m}{a^n} \)
  • Solution: Subtract the exponents: \( a^{m-n} \)
This rule simplifies computations involving powers and is used in many areas of algebra. By following this rule, you can effortlessly solve problems like \( \frac{a^8}{a^8} \), which reduces to \( a^{8-8} = a^0 \).
Exponent Rules
Exponent rules are essential building blocks in algebra, guiding how we perform operations involving powers. Let's look at some of these rules briefly:
  • Product of Powers: To multiply powers with the same base, add their exponents: \( a^m \times a^n = a^{m+n} \).
  • Quotient of Powers: As previously mentioned, subtract the exponent of the denominator from that of the numerator: \( \frac{a^m}{a^n} = a^{m-n} \).
  • Power of a Power: To raise a power to another power, multiply the exponents: \( (a^m)^n = a^{m\cdot n} \).
  • Zero Exponent Rule: Any non-zero base raised to the power of zero is 1: \( a^0 = 1 \).
Each of these rules plays a crucial role when simplifying expressions and solving algebraic problems. Mastering them will make handling complex equations much easier.
Zero Exponent Rule
The zero exponent rule is an intriguing concept in the world of exponents. It states that any non-zero number raised to the zero power equals 1. This might seem puzzling at first, but it is logical when you consider the pattern formed by dividing powers:Let's look at a practical example. Consider powers of 2 under division:
  • \( \frac{2^3}{2^3} = 2^{3-3} = 2^0 \)
  • But \( \frac{2^3}{2^3} = 1 \), so \( 2^0 = 1 \)
This pattern holds true for all non-zero numbers. The zero exponent rule complements the other exponent rules by allowing us to simplify expressions, ensuring that computations remain consistent and straightforward. Understanding this rule will reinforce your grasp of exponents and enhance your ability to solve algebraic problems efficiently.

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Most popular questions from this chapter

Numbers can also be expressed in expanded form. Example \(1: 13,548=10,000+3000+500+40+8\) \(=\left(1 \times 10^{4}\right)+\left(3 \times 10^{3}\right)+\left(5 \times 10^{2}\right)+\left(4 \times 10^{1}\right)+\left(8 \times 10^{0}\right)\) Example \(2: 0.568=0.5+0.06+0.008\) $$ =\left(5 \times 10^{-1}\right)+\left(6 \times 10^{-2}\right)+\left(8 \times 10^{-3}\right) $$ Write each number in expanded form. $$0.173$$

CHALLENGE Is it true that \(\frac{23}{53}=\frac{23}{53}\) or \(\frac{2}{5} ?\) Explain why or why not.

In a pen factory, an average of 5 pens out of every \(p\) pens tested are rejected. What fraction of those \(p\) pens is NOT rejected? $$\mathbf{F} \frac{5}{p}$$ $$G \frac{p}{5}$$ $$H \frac{p-5}{5}$$ $$J \frac{p-5}{p}$$

Find each product or quotient. Express using exponents. $$4^{4} \cdot 4^{7}$$

Find each product. $$5.0 \times 0.0001$$
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