Problem 56 - Algebra 2

Question

Simplify. Assume that no variable equals 0 \(\left(x^{3}\right)^{4}\)

Step-by-Step Solution

Answer

The simplified expression is \(x^{12}\).

Step 1: Identify the Exponential Rule

In this problem, we need to simplify an expression where an exponent is raised to another exponent, specifically \((x^3)^4\). To handle this, we apply the power of a power rule which states that \((a^b)^c = a^{b imes c}\).

Step 2: Apply the Power of a Power Rule

Use the rule identified in Step 1 to rewrite the expression. Here the base is \(x\), the first exponent is 3, and the second exponent is 4. According to the rule, multiply these exponents. So, \((x^3)^4 = x^{3 imes 4}\).

Step 3: Simplify the Exponents

Perform the multiplication from Step 2 to find a single exponent. Multiply 3 by 4 to get 12. Therefore, the expression simplifies to \(x^{12}\).

Key Concepts

Exponential Expressions

Exponential expressions are mathematical expressions where a base number is raised to a power, known as an exponent. They are commonly written in the format of \( a^b \), where \( a \) is the base, and \( b \) is the exponent. Exponents provide a shorthand way to express repeated multiplication. For example, \( x^3 \) means \( x \) multiplied by itself three times: \( x \cdot x \cdot x \).
When you encounter expressions like \((x^3)^4\), you're dealing with an exponent raised to another exponent. This is called a "power of a power". It's important to understand how to manage such expressions, as they can be simplified using specific rules. These rules help in turning a complex expression into a simpler form, making calculations easier and more efficient.

Simplifying Exponents

Simplifying exponents involves reducing expressions with exponents to their simplest form. One crucial rule in this process is the power of a power rule. When an exponent is raised to another exponent, such as in \((a^m)^n\), the expression can be simplified by multiplying the exponents: \(a^{m \times n}\). This is what you do in the given exercise to simplify \((x^3)^4\) into \(x^{12}\).
Other rules include:

• Product of Powers Rule: \(a^m \cdot a^n = a^{m+n}\). You add the exponents when multiplying like bases.
• Power of a Product Rule: \((ab)^n = a^n \cdot b^n\). You distribute the exponent to both bases inside the parentheses.

Understanding these rules helps in recognizing patterns and applying them to simplify various exponential expressions efficiently.

Multiplying Exponents

When multiplying exponents, particularly in cases of a "power of a power" like \((x^3)^4\), the exponents are multiplied together. This is distinct from other operations of exponents but crucial for simplifying them further. In the given problem, you start with \((x^3)^4\), and through the power of a power rule, this becomes \(x^{3 \times 4}\), resulting in \(x^{12}\).
Remember:

• Power of a Power: Multiply the exponents.
• Product of Powers: When multiplying two terms with the same base, add the exponents.
• Always ensure the base stays the same; only the exponents are altered during multiplication, addition, or simplification.

By consistently applying these rules, you can navigate and simplify complex exponential expressions with confidence.