Problem 23 - Algebra 1: Concepts and Skills

Question

Copy and complete the statement. $$ \frac{a^{10}}{a^{4}}=a^{?} $$

Step-by-Step Solution

Answer

The result of the operation \(a^{10}/a^{4}\) is \(a^{6}\).

Step 1: Identify the Law of Exponents to be Applied

In this case, the law of exponents for division must be applied. This law states that \(a^{m}/a^{n} = a^{m-n}\) where a is the base, m is the divisor exponent and n is the dividend exponent.

Step 2: Apply the Law of Exponents

Applying this law to our problem, \(a^{10}/a^{4}\), we subtract the exponent of the denominator from the exponent of the numerator: \(10 - 4 = 6\).

Key Concepts

Exponent Division Rule

Understanding the exponent division rule is essential when working with exponential expressions. When dividing two exponents with the same base, you subtract the exponent of the denominator from the exponent of the numerator. The mathematical representation of this rule is \( \frac{a^m}{a^n} = a^{m-n} \), where \(a\) is a non-zero base, \(m\) is the exponent of the numerator, and \(n\) is the exponent of the denominator. We call this the quotient of powers property. For example, if we take \( \frac{a^{10}}{a^4} \), following this rule, we would subtract 4 from 10, giving us \( a^6 \).

• It simplifies complex expressions.
• Only exponents with the same base can be divided in this way.
• The base \(a\) must not be zero, as the division by zero is undefined.

Remember, this applies to any base with a real number exponent, so it's a broadly useful rule in algebra.

Exponential Expressions

Exponential expressions are a way to express repeated multiplication of the same number. They consist of a base and an exponent. The base is the number being multiplied, and the exponent, a positive or negative integer, shows how many times the base is used as a factor. For instance, \(b^3\) means you multiply \(b\) by itself three times: \(b \times b \times b\).To work effectively with exponential expressions, remember these key points:

• The base can be any real number.
• The exponent tells us the number of times the base is multiplied by itself.
• Exponential expressions follow specific laws, such as the exponent division rule and the rules for multiplying and raising a power to a power.

A good grasp of how exponential expressions work will enable you to handle a wide range of mathematical problems, including those involving exponential growth and decay.

Simplifying Exponents

Simplifying exponents is a way to make them more manageable and can be done using the laws of exponents. These laws include the quotient (division), product (multiplication), power (raising an exponent to another power), and zero exponent rules, among others. When simplifying, you follow these rules systematically to reduce expressions to their simplest form. For simplified expressions, the aim is to have as few terms as possible and the smallest exponent possible.For example:

• Use the exponent division rule when dividing like bases.
• Multiply first when you have the same base and you are multiplying the exponents.
• If you have an exponent raised to another exponent (\((a^m)^n\)), you multiply the exponents (\(a^{m \times n}\)).
• A base with an exponent of zero is always one (\(a^0 = 1\)).

In the exercise \( \frac{a^{10}}{a^4} \), you simplify the expression by subtracting the lower exponent from the higher one, resulting in \(a^6\). This simplification not only makes calculations easier but also helps in better understanding the underlying mathematical concepts.