Problem 26 - Algebra 1: Concepts and Skills

Question

Simplify the quotient. $$ \frac{8^{2}}{8^{3}} $$

Step-by-Step Solution

Answer

The simplest form of the quotient is \(8^{-1}\).

Step 1: Understand the Law of Exponents for Division

In an equation of the form \(a^{n}/a^{m}\), we can simplify this by applying the law of exponents for division. The law states that when you divide with the same base, you subtract the exponent of the denominator from the exponent of the numerator.

Step 2: Apply the Law of Exponents for Division

Using the law from Step 1, this equation simplifies to \(8^{(2-3)}\).

Step 3: Simplify the Operation

Solving the operation in the parentheses we get \(8^{-1}\).

Key Concepts

Law of Exponents

Exponents are the shorthand way of expressing repeated multiplication of the same number. The base is the number that is repeatedly multiplied, and the exponent tells how many times the multiplication occurs. There are several laws around exponents that make working with them much simpler. One fundamental rule you need to know is the law of exponents for division, which is especially useful in simplifying expressions.

The law of exponents for division states: If you have two exponents with the same base, and you are dividing them, you can simply subtract the exponent of the denominator from the exponent of the numerator. This rule is written as:

• \(a^n \div a^m = a^{n-m}\)

Understanding this principle allows you to transform complex expressions into more manageable ones through subtraction instead of division.

By becoming familiar with this law, you ensure faster, more efficient calculations when simplifying expressions that involve exponents.

Simplifying Expressions

Simplifying mathematical expressions is about making them easier to understand or work with without changing their value. When it comes to expressions involving exponents, simplification often involves using the laws of exponents to reduce the complexity of the terms.

Take the example of simplifying \( \frac{8^{2}}{8^{3}} \). Using the law of exponents for division, the expression can be reduced quickly:

• Subtract the exponents: \(2 - 3\)
• This gives you \(8^{-1}\)

In many situations, further simplification can also mean rewriting the expression in a different form, such as using fractional notation or getting rid of negative exponents. For instance, \(8^{-1}\) can be written as \(\frac{1}{8}\).

Mastering the art of simplification helps in solving equations efficiently and is a vital skill in algebra and beyond.

Division of Exponents

Division of exponents often appears daunting at first, but it's straightforward once you understand the rules. When confronted with the division of exponential terms that share the same base, you turn to the division rule of exponents. This rule is a powerful tool that tells you that instead of actually dividing the terms, you subtract the exponent of the lower term from the upper term's exponent.

In our example of \( \frac{8^{2}}{8^{3}} \), instead of performing complex division, you use subtraction:

• Take the numerator's exponent \(2\) and subtract the denominator's exponent \(3\).
• The result is \(8^{-1}\), which is much simpler.

This result, \(8^{-1}\), signals that you have successfully divided the exponents, and it can be interpreted as \(\frac{1}{8}\), providing a clearer numeric value.

Grasping division of exponents transforms your approach to mathematical problems, offering a pathway to simplify otherwise complicated calculations.