Problem 19 - Algebra 1: Concepts and Skills

Question

Copy and complete the statement. $$ \frac{3^{9}}{3^{5}}=3^{?} $$

Step-by-Step Solution

Answer

The result of the statement, \(\frac{3^{9}}{3^{5}} = 3^{?}\), is 4

Step 1: Define the Rule

Remember that when you divide two numbers that have the same base, you subtract the powers. Symbolically, this can be written as \(a^{m}/a^{n} = a^{m-n}\).

Step 2: Apply the Rule

With 3 as the common base and 9 and 5 as the powers in the problem, the rule dictates that \(3^{9}/3^{5} = 3^{9-5}\).

Step 3: Subtract Powers

Subtract the powers to get \(3^{9-5} = 3^{4}\).

Key Concepts

Division of Powers

When dealing with division of powers, it essentially involves dividing numbers that have the same base. This process is straightforward due to the fundamental rule of exponents: when you divide two powers with the same base, you subtract the exponents. For example, in the exercise \[ \frac{3^9}{3^5} \]the base number is 3 in both the numerator and the denominator. This means we can apply the exponent division rule.

• The key is maintaining the base as it is.
• Then, simply subtract the exponent in the denominator from the exponent in the numerator.

Ultimately, division of powers is nothing more than a simplification by subtraction, making calculations more efficient and error-proof. The concept helps in reducing complex equations to simpler expressions which align with arithmetic laws.

Base and Exponent

Understanding the components "base" and "exponent" is crucial when working with powers. The base is the repeating factor in multiplication, while the exponent tells us how many times the base is used as a factor. For example, in \[ 3^9 \]3 is the base and 9 is the exponent. This means the calculation represents 3 multiplied by itself 9 times.

• The base can be any real number.
• Exponents can be positive, negative, or even zero, each altering the multiplication's outcome.

Having a firm grasp on what bases and exponents represent allows for easier manipulation of expressions, including multiplying, dividing, and simplifying powers. The base remains the same in operations like division of powers, while the exponents are adjusted according to the operation's specific rule.

Subtraction of Exponents

The subtraction of exponents is a simple yet important concept when dividing powers with the same base. This operation allows us to reduce complex expressions into simpler forms. According to the division of powers rule, when you divide numbers with the same base, you subtract the exponent in the denominator from that in the numerator. In the exercise \[ \frac{3^9}{3^5} \]we subtract 5 from 9, resulting in 4. Therefore, \[ 3^{9-5} = 3^4 \].

• Ensures that the base stays the same.
• Makes significant simplifications possible.

This rule is simple but essential, especially in algebra and higher-level mathematics, where equations often involve powers. By practicing subtraction of exponents, students can quickly and accurately simplify exponential expressions.