Problem 1 - Algebra 1: Concepts and Skills

Question

Match the division property of exponents with the example that illustrates it. A. \(\left(\frac{3}{6}\right)^{2}=\frac{3^{2}}{6^{2}}\) B. \(\frac{4^{3}}{4^{5}}=4^{3-5}\) Quotient of powers property

Step-by-Step Solution

Answer

Example B illustrates the division property of exponents, also known as the quotient of powers property.

Step 1: Understanding the Division Property of Exponents

First, let's understand the division property of exponents, also known as the quotient of powers property. This property states that for any non-zero number 'a' and integers 'm' and 'n', \(a^{m}/a^{n} = a^{m-n}\). This property suggests that if we're dividing two exponents with the same base, we subtract the exponent of the denominator from the exponent of the numerator.

Step 2: Analyzing Example A

In Example A, the expression \(\left(\frac{3}{6}\right)^{2}\) is given, which simplifies to \(\frac{3^{2}}{6^{2}}\). However, this does not match the form of the quotient of powers property, thus, the division property of exponents does not apply here.

Step 3: Analyzing Example B

In Example B, the expression \(\frac{4^{3}}{4^{5}}=4^{3-5}\) is given. This is exactly the form of the quotient of powers property where the bases (4 in this case) are equal and the exponent of the numerator (3) is subtracted from the exponent of the denominator (5). Hence, Example B illustrates the quotient of powers property.

Key Concepts

Quotient of Powers Property

The "Quotient of Powers Property" is a key concept when working with exponents. It simplifies the division of two expressions that have the same base but different exponents. This property tells us that when dividing powers with the same base, we should subtract the exponent in the denominator from the exponent in the numerator.

This is mathematically expressed as: if we have a base 'a', and two exponents 'm' and 'n', then \[\frac{a^m}{a^n} = a^{m-n}\]

This property is particularly useful because it reduces complex expressions into simpler forms. For example, if you encounter an expression like \( \frac{4^3}{4^5} \), using the quotient of powers property, it can be simplified to \( 4^{3-5} = 4^{-2} \).

Remember, you can only apply this property when the bases are the same. In the provided exercise, Example B:\( \frac{4^3}{4^5} = 4^{3-5} \) is a perfect demonstration of this rule.

Exponent Laws

Exponent laws are rules that help us handle powers and make calculations involving exponents simpler. These laws apply to multiplying, dividing, and raising powers to another power. They are essential for solving problems that involve exponential expressions.

Here are some key exponent laws to remember:

• Product of Powers Law: When multiplying two exponents with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \)
• Quotient of Powers Law: When dividing two exponents with the same base, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a Power Law: When raising an exponent to another power, you multiply the exponents: \((a^m)^n = a^{m \cdot n} \)

Exponent laws provide a foundation for you to simplify and solve problems involving powers, ensuring calculations are accurate and efficient.

Subtraction of Exponents

The "Subtraction of Exponents" concept is part of the division process when applying the quotient of powers property. It explains how and why we subtract exponents when dividing similar bases.

When you have a division situation where the bases are the same, such as \( \frac{a^m}{a^n} \), you subtract the exponent in the denominator, 'n', from the exponent in the numerator, 'm'. This logical step reduces your expression to a simpler form, which is \( a^{m-n} \).

For example, consider the expression \( \frac{4^3}{4^5} \). Applying the subtraction of exponents step, we calculate \( 4^{3-5} = 4^{-2} \). This simplification can help you better understand and handle more complex algebraic expressions by ensuring they are in their simplest forms.