Problem 25 - Algebra 1: Concepts and Skills

Question

Copy and complete the statement. \(\left(5^{?}\right)^{3}=5^{9}\)

Step-by-Step Solution

Answer

? = 3

Step 1: Identify the relationship between the bases and exponents

Since the bases on both sides of the equation are identical, the values of the exponents must also be equal. Thus, \(3*? = 9\)

Step 2: Solve for the unknown

To find the value of '?', we divide 9 by 3. Thus, ? = 9/3

Key Concepts

Algebraic Equations

In the world of mathematics, algebraic equations are like puzzles waiting to be solved. An algebraic equation is a mathematical statement where two expressions are set equal to each other with one or more variables.
The goal? To find a numerical value that makes the equation true. Consider the equation \(3x + 2 = 11\). Here, you want to find the value of \(x\) that makes both sides of the equation equal.

• Equations can often be manipulated by applying various operations such as addition, subtraction, multiplication, and division on both sides.
• To solve these puzzles, understand that what you do to one side, you must do to the other. This keeps the equation balanced, just like a perfectly balanced seesaw.

Algebraic equations come in different forms, including linear, quadratic, and polynomial equations. The process of solving these equations often involves isolating the variable. This is where equality comes into play.
Remember, the equations express equality and help determine the relationship between the variables.

Properties of Exponents

Exponents are handy tools in mathematics that simplify the process of expressing repeated multiplication. Instead of writing \(5 \times 5 \times 5\), you can write \(5^3\). Here, \(5\) is the base, and \(3\) is the exponent or power.

• The property \((a^m)^n = a^{m \times n}\) states that when raising a power to another power, you multiply the exponents.
• Another useful property is \(a^m \cdot a^n = a^{m+n}\), which comes into play when multiplying like bases.

Understanding these properties allows you to simplify complex expressions and solve equations much easier. For example, in the original exercise, \((5^?)^3 = 5^9\) showcases the exponent power rule. Solving such problems requires mastering these properties, which are the building blocks of algebra.

Solving Equations

When solving equations involving exponents, it's crucial to apply the right properties to simplify terms. The ultimate goal is to isolate the variable or unknown.
Here’s how you can tackle such problems:- Start by checking if there are similar bases on both sides of the equation, as in the exercise \((5^?)^3 = 5^9\).- Use the necessary exponent rules, for instance, multiplying exponents when there’s a "power of a power."- If the base is the same, make the exponents equal to each other, for example, \(3 \times ? = 9\).- Solve the resulting simple equation to get the value of the unknown, here: \(? = 9/3 = 3\).Applying these steps helps demystify even the toughest equations. Solving equations is like unwinding a mystery by logically following through with operations that make the puzzle's complexities disappear step-by-step.