Problem 61 - Algebra 2

Question

\(\mathrm{ACT} / \mathrm{SAT}\) If \(3^{5} \cdot p=3^{3},\) then \(p=\) $$ \begin{array}{l}{\mathrm{A}-3^{2}} \\ {\mathrm{B} 3^{-2}} \\ {\mathrm{C} \frac{1}{3}} \\ {\mathrm{D} 3^{\frac{1}{3}}}\end{array} $$

Step-by-Step Solution

Answer

The answer is B: \(3^{-2}\).

Step 1: Understand the Equation

The equation given is \(3^5 \cdot p = 3^3\). Our task is to solve for \(p\).

Step 2: Isolate \(p\)

To isolate \(p\), divide both sides of the equation by \(3^5\). This yields \(p = \frac{3^3}{3^5}\).

Step 3: Simplify the Fraction

The fraction \(\frac{3^3}{3^5}\) can be simplified using the property of exponents, \(\frac{a^m}{a^n} = a^{m-n}\). Hence, \(\frac{3^3}{3^5} = 3^{3-5}\).

Step 4: Evaluate the Exponent

Compute \(3^{3-5}\), which results in \(3^{-2}\).

Step 5: Match with Given Options

The expression \(3^{-2}\) corresponds to option (B) in the given choices.

Key Concepts

Understanding Exponents

Exponents play a crucial role in mathematics, especially when simplifying expressions and solving equations. An exponent indicates how many times a number, known as the base, is multiplied by itself. For instance, in the expression \(3^5\), three is the base, and five is the exponent, meaning \(3\) is multiplied by itself five times: \(3 \times 3 \times 3 \times 3 \times 3\).

When working with equations that involve exponents, understanding the laws or rules of exponents can greatly simplify the process. The key rules include:

• Multiplication of like bases: \(a^m \cdot a^n = a^{m+n}\)
• Division of like bases: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a power: \((a^m)^n = a^{m \cdot n}\)

By applying these rules, you can simplify expressions to help find solutions. In the given problem, the rule for division of exponents is used, where subtracting the exponents simplifies the expression \(\frac{3^3}{3^5}\) to \(3^{3-5}\).

Solving Equations

Solving equations is about finding the value of the unknown variable that satisfies the equation. When solving, the goal is typically to isolate the variable on one side of the equation. This involves performing operations that maintain the balance of the equation, such as addition, subtraction, multiplication, or division.

In the problem presented, the equation \(3^5 \cdot p = 3^3\) involves isolating \(p\). To achieve this, you divide both sides by \(3^5\) to shift all terms involving the base \(3\) from the side with \(p\). This operation helps to show the equation in terms of \(p\), leading to \(p = \frac{3^3}{3^5}\).

By isolating \(p\), the next step is to use the properties of exponents to further simplify and determine the value of \(p\). This demonstrates the importance of understanding and applying algebraic principles when working through mathematical solutions.

Simplifying Expressions

Simplifying expressions means reducing them to their most basic form, making them easier to understand or compare. While simplifying, the properties of numbers and exponents are key, as these laws allow us to rewrite expressions in a more digestible form.

For example, the fraction \(\frac{3^3}{3^5}\) can be simplified by utilizing the property of exponents that lets us subtract the exponents of identical bases. This is visually simplified to \(3^{3-5}\), which calculates to \(3^{-2}\).

The result \(3^{-2}\) is itself a simplified expression representing the original fraction. It is an example of working with negative exponents, where \(3^{-2}\) equates to \(\frac{1}{3^2}\) or \(\frac{1}{9}\). Simplifying such expressions is pivotal for solving equations quickly and effectively.