Problem 117 - Algebra A Combined Function

Question

Fill in the boxes so that each statement is true. $$ x^{\square} \cdot x^{\square}=x^{12} $$

Step-by-Step Solution

Answer

Each box should contain the number 6.

Step 1: Understand the Property of Exponents

When multiplying powers with the same base, you add the exponents. The property can be written as \( x^a \cdot x^b = x^{a+b} \). We will use this property to solve the given exercise.

Step 2: Express the Given Equation

We start with the equation from the exercise: \( x^{\square} \cdot x^{\square} = x^{12} \). This implies that both exponents \( \square \) are the same.

Step 3: Set the Exponents Equal

Since the bases are the same, we can equate the exponents while using the property from Step 1. Therefore, \( a + a = 12 \) where \( a \) is the unknown exponent.

Step 4: Solve for the Unknown Exponent

From the equation \( a + a = 12 \), simplify the left side to get \( 2a = 12 \). Now, solve for \( a \) by dividing both sides by 2, yielding \( a = 6 \).

Step 5: Verify the Solution

Substitute the solved exponent back into the original context to ensure correctness: \( x^6 \cdot x^6 = x^{6+6} = x^{12} \). The calculation is correct and verifies that the solution is consistent.

Key Concepts

Multiplication of Powers

Multiplying powers is a crucial concept in algebra, especially when dealing with similar bases. When you multiply powers that have the same base, you must consider their exponents. This leads us to the essential property: if you have two powers like \( x^a \) and \( x^b \), multiplying them results in \( x^{a+b} \). This is because the operation effectively combines the total number of factors expressed by their exponents.
For instance, if \( x^3 \) is multiplied by \( x^4 \), you add the exponents (since there are three factors of \( x \) in the first term and four in the second), giving \( x^7 \) as the result. Understanding this combination process helps make more complex algebraic expressions easier to manage.

Exponents

Exponents, sometimes referred to as powers or indices, indicate how many times a number (the base) is multiplied by itself. In the expression \( x^n \), \( x \) is the base and \( n \) is the exponent. Exponents are a shorthand way to express repeated multiplication.

• \( x^1 \) represents the base itself.
• \( x^0 \) equals 1 for any nonzero base, which might be counterintuitive but is rooted in the properties of exponents during division.
• Negative exponents inversely express powers (\( x^{-n} = \frac{1}{x^n} \)).

Mastering the basic rules of exponents allows for simplified manipulation of equations and expressions in mathematics.

Mathematical Equations

Equations are mathematical statements that assert the equality of two expressions. An equation contains one or more unknown variables that you need to solve for. In the exercise, the equation is \( x^{\square} \cdot x^{\square} = x^{12} \). Here, the task is to determine the value of the unknown exponent represented by the square icon.
The solving process involves replacing the complex side of an equation with a simpler one by using laws of algebra, such as combining like terms, using properties of exponents, and following operations like addition or subtraction. By setting the equation \( a + a = 12 \), you solve for \( a \) by isolating it, often through basic operations such as division or multiplication.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and operations combined but without an equality sign, unlike an equation. They are fundamental in formulating mathematical models and representing real-world problems. In the context of the exercise, knowing how to work with expressions like \( x^a \cdot x^b \) is vital for simplifying and transforming equations.
The manipulation of algebraic expressions often involves:

• Applying the distributive property.
• Combining like terms.
• Simplifying using exponent laws.

Mastering how to handle manipulations builds a strong foundation for solving complex mathematical problems, as these skills are recursively applied when solving equations. Understanding the interconnections between expressions and equations is essential for a deeper comprehension of algebra.