Problem 91 - Algebra 2

Question

What is the ratio of \(\left(65 \cdot 3^{17}\right)\) to \(\left(65 \cdot 3^{14}\right) ?\) $$ \begin{array}{lllll}{\text { A. } 17 \text { to } 14} & {\text { B. } 3 \text { to } 1} & {\text { C. } 9 \text { to } 1} & {\text { D. } 27 \text { to } 1}\end{array} $$

Step-by-Step Solution

Answer

Therefore, the ratio of \((65 \cdot 3^{17})\) to \((65 \cdot 3^{14})\) is 27 to 1. So, the answer is D. 27 to 1.

Step 1: Break Down the Problem

The problem asks for the ratio of \((65 \cdot 3^{17})\) to \((65 \cdot 3^{14})\). Notice that both quantities have a common factor, which is 65, and the difference in the powers of 3. The ratio can be simplified by eliminating the common factor and only focusing on the powers of 3.

Step 2: Simplify the Expressions

We can simplify the ratio by using the properties of exponents. The expression thus becomes \((3^{17}) / (3^{14})\) This simplifies to \(3^{(17-14)}\), or \(3^3\).

Step 3: Calculate the Simplified Expression

Calculate the expression \(3^3\) which equals to 27.

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing complex mathematical expressions to their simplest form. It often makes solving problems quicker and easier. In this problem, both \(65 \cdot 3^{17}\) and \(65 \cdot 3^{14}\) have a common factor. In math, a common factor is a value that divides each term in an expression equally. Here, the number 65 is common in both, so you can factor it out. This helps in simplifying the expression further. The core action here is to observe such commonalities in mathematical problems. By dividing each term by 65, we focus on the remaining part of the expression: the powers of 3. This makes the problem more manageable.

Ratio and Proportion

Ratios and proportions are a way to compare quantities. A ratio expresses the magnitude of one quantity relative to another. In this exercise, you're asked to find the ratio of the expression \(65 \cdot 3^{17}\) to \(65 \cdot 3^{14}\). Here, you determine how many times one number contains another. Ratios can be simplified when both numbers share a common multiple or factor. This means we can simplify things like fractions.For the given expressions, by simplifying, you focus on the result of dividing one by the other. When you simplify \(3^{17} \/ 3^{14}\), you actually find the ratio between these two expressions. The resulting simplified form, 27, shows that the expression \(65 \cdot 3^{17}\) is 27 times larger than \(65 \cdot 3^{14}\). This ratio simply indicates the proportionate difference between the two expressions.

Properties of Exponents

Properties of exponents are rules that simplify expressions involving exponents. An exponent indicates how many times a number, known as the base, is multiplied by itself. In this problem, we focus on properties such as the division of like bases. The property states:

• To divide powers with the same base, subtract the exponents: \(a^m \/ a^n = a^{m-n}\).
• Any number raised to the power of 0 is 1 (like \(a^0 = 1\)).

Using these properties, the problem \(3^{17} \/ 3^{14}\) simplifies to \(3^{3}\). This property allows quick calculation by reducing complex terms. It highlights the efficiency and power of using exponential rules in simplifying mathematical expressions. The exponent property strategically resolves the problem with fewer steps, making calculations more efficient.