Problem 118 - Algebra A Combined Function

Question

Fill in the boxes so that each statement is true. $$ \left(x^{\square}\right)^{\square}=x^{20} $$

Step-by-Step Solution

Answer

Use exponents 4 and 5: \((x^4)^5 = x^{20}\).

Step 1: Understanding the expression

The expression given is \( \left(x^{\square}\right)^{\square}=x^{20} \). This follows the power of a power rule in exponents, which states that \( \left(a^m\right)^n = a^{m \times n} \). We need to find two numbers that multiply together to give 20.

Step 2: Identifying the base power

We are working with the equation \( \left(x^{a}\right)^{b}=x^{20} \), meaning \( a \times b = 20 \). We must choose values for \( a \) and \( b \) such that their product is 20.

Step 3: Breaking down 20 into factors

Consider the factor pairs of 20. These pairs include (1, 20), (2, 10), and (4, 5). Any of these pairs can be potential values for \( a \) and \( b \).

Step 4: Selecting a pair of exponents

Select a valid pair. For simplicity, choose \( a = 4 \) and \( b = 5 \). This means \( \left(x^{4}\right)^{5}=x^{20}\), which conforms to the original equation.

Key Concepts

Power of a Power Rule

The expression \( \left(x^{\square}\right)^{\square}=x^{20} \) provides an excellent opportunity to explore the **power of a power rule** in exponents. This mathematical rule is fundamental when dealing with exponents raised to another power. Here's the core idea:

• When you raise a power to another power, you simply multiply the exponents.
• The formula is written as \( \left(a^m\right)^n = a^{m \times n} \).

In our exercise, this means identifying two numbers \( a \) and \( b \) such that \( a \times b \) equals 20, given that \( \left(x^a\right)^b = x^{20} \). This approach simplifies complex exponential expressions, breaking them into manageable parts. To solve, we factor 20 into a product of two numbers.

Factorization

To apply the power of a power rule effectively, we first need to **factorize** the number 20. Factorization is the process of breaking down a number into its multiplicative components or factors.

• It involves finding pairs of numbers which, when multiplied together, yield the original number (in this case, 20).
• The factor pairs of 20 are (1, 20), (2, 10), and (4, 5).

This is crucial as it allows us to choose pairs that multiply to 20, informing us which exponents to use in the power expression \( \left(x^a\right)^b \). In this context, selecting the pair (4, 5), for example, transforms \( \left(x^4\right)^5 \) to the simpler form \( x^{20} \), maintaining the equation's integrity.

Algebraic Expressions

An **algebraic expression** like \( \left(x^{\square}\right)^{\square}=x^{20} \) consists of numbers, variables, and exponents. In algebra, these symbols express mathematical relationships and operations. Grasping the structure of such expressions is key to manipulating and solving them.

• An algebraic expression becomes an equation when it is set equal to another expression or number.
• In our problem, the expressions involve powers and exponents, characteristics typical of algebraic simplification tasks.

Understanding the components and how they relate to each other allows us to apply algebraic rules, like the power of a power rule, efficiently—resulting in a neatly solved equation.

Math Problem Solving

**Math problem solving** involves a systematic approach to find solutions to mathematical questions or exercises, like our exponent equation. This methodical process is built on logical thinking and understanding mathematical principles.

• Start by comprehending the problem through careful reading and identifying all given information and required outcomes.
• Incorporate learned rules and factors, like the power of a power rule, and factorization to manage complex expressions logically.
• Trial and error can be useful in situations like choosing appropriate factor pairs.

This organized approach empowers students to handle algebraic challenges. By applying strategies such as breaking complex problems into simpler parts and methodically testing solutions, we achieve an accurate and efficient resolution.