Problem 36 - Algebra 1: Concepts and Skills

Question

Write the expression as a single power of the base. \(\left(x^{3}\right)^{2}\)

Step-by-Step Solution

Answer

The simplified expression as a single power of the base is \(x^{6}\)

Step 1: Understanding the Problem

The expression given is \((x^{3})^{2}\). This is an example of a power of a power in exponentiation, which means to multiply the powers.

Step 2: Applying Power of a Power Rule

Using the power of a power rule, \((x^{3})^{2} = x^{3*2}\)

Step 3: Final Calculation

Multiplying the powers 3 and 2, we get \(x^{6}\)

Key Concepts

Power of a Power Rule

Exponentiation involves raising a number or variable to a certain power. When an exponent is itself raised to another power, we use the power of a power rule. This rule is key to simplifying expressions like \((x^3)^2\). Here’s how it works:

• Take the base, which in our case is \(x\).
• Multiply the inner exponent, \(3\), by the outer exponent, \(2\).
• The result becomes \(x^{3 \times 2} = x^6\).

This simplification helps in solving equations more efficiently by reducing multiple layers of exponents into one. Keep practicing this rule to master handling complex expressions in algebra.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition and multiplication). In the expression \((x^3)^2\), we are focusing on how exponents function within these expressions.

Here’s what you need to know:

• The base, \(x\), can be any number or variable.
• Exponents show how many times to multiply the base by itself.
• In algebra, combining exponents efficiently is crucial for simplification.

Understanding algebraic expressions and their components can make learning easier, especially when dealing with exponential expressions.

Multiplying Exponents

When exponents are involved, knowing how to multiply them is essential. Consider the expression \((x^3)^2\). Let’s break down the process:

• Apply the power of a power rule by multiplying the exponents: \(3 \times 2\).
• Always keep the base the same, here it’s \(x\).
• Place the new exponent above the base: \(x^6\).

This rule means you can swiftly transform expressions like \((x^3)^2\) into simpler forms like \(x^6\). Practicing this enables you to tackle more challenging problems with confidence.