Problem 2 - 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. \(\frac{4^{3}}{4^{5}}=4^{3-5}\) Power of a quotient property

Step-by-Step Solution

Answer

Example 'A' illustrates the Power of a Quotient Property.

Step 1: Understand the Power of a Quotient Property

The Power of a Quotient Property explains that when a fraction is raised to an exponent, each part of the fraction (the numerator and the denominator) is raised to that exponent independently. It can be expressed as \((a/b)^n = a^n / b^n\)

Step 2: Match the Property with examples

Now, match the property with the provided examples. Look for an example where a fraction is raised to a power, and that power is applied to both the numerator and the denominator. When we look at the first example A, \(\left(\frac{3}{6}\right)^{2}=\frac{3^{2}}{6^{2}}\), we can see that this follows the Power of a Quotient rule. In the second example B, \(\frac{4^{3}}{4^{5}}=4^{3-5}\), we are not dealing with a single fraction raised to an exponent, but rather this illustrates a different rule of exponentiation known as the quotient of powers property, which states that when dividing like bases, subtract the exponents.

Step 3: Conclusion

Thus, match the Power of a Quotient Property with example A, because it best illustrates the rule outlined by the Power of a Quotient Property.

Key Concepts

Power of a Quotient

The power of a quotient property is a fascinating concept in mathematics. It helps us understand how to handle situations when we raise an entire fraction to a power. When we have a fraction, say \( \left( \frac{a}{b} \right)^n \), the property tells us that each part of the fraction, the numerator and the denominator, is raised to the power individually. This turns into \( \frac{a^n}{b^n} \).
For a better grasp, imagine having \( \left( \frac{3}{6} \right)^2 \). Applying the power of a quotient property, you need to raise both 3 and 6 to the power of 2. This results in \( \frac{3^2}{6^2} \). It simplifies the calculation and makes it manageable.

Quotient of Powers Property

The quotient of powers property comes into play when dividing similar bases raised to different exponents. In mathematical terms, if we have \( \frac{a^m}{a^n} \), we subtract the exponent of the denominator from the exponent of the numerator. This results in \( a^{m-n} \).
Consider the example \( \frac{4^3}{4^5} \). Here, the base 4 is the same, so we subtract the exponents: \( 3 - 5 \). This simplifies to \( 4^{-2} \), indicating how the exponents directly affect the calculations.
This property is particularly useful in simplifying expressions and solving equations involving powers and exponents.

Exponents Rules

Understanding exponents is crucial as they are used widely in algebra and higher-level math. There are several key rules to remember:

• Power of a Power: To raise a power to another power, multiply the exponents. For example, \((a^m)^n = a^{m\cdot n}\).
• Product of Powers: When multiplying like bases, add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• Zero Exponent Rule: Any non-zero base raised to the zero power is 1: \(a^0 = 1\).
• Negative Exponent Rule: A negative exponent means the reciprocal of the base is taken to the positive exponent: \(a^{-n} = \frac{1}{a^n}\).

By mastering these basic exponent rules, you can solve many complex mathematical problems more easily. They simplify tasks and empower you to manage exponential functions and equations effectively.