Problem 31 - College Algebra

Question

Simplify each exponential expression $$ \left(x^{3}\right)^{7} $$

Step-by-Step Solution

Answer

The simplified form of \( (x^{3})^{7} \) is \( x^{21} \)

Step 1 - Identify the Base and the Exponents

First, identify the base which is \( x \), and the exponents which are \( 3 \) and \( 7 \). We have \( (x^{3})^{7} \), meaning \( a = x \), \( m = 3 \), and \( n = 7 \).

Step 2 - Apply the Power of a Power Rule

Then, apply the power of a power rule, which states that \( (a^{m})^{n} = a^{m*n} \). In this case, the rule will give us: \( (x^{3})^{7} = x^{3*7} \)

Step 3 - Multiply the Exponents

Finally, carry out the multiplication in the exponent to simplify. We have: \( x^{3*7} = x^{21} \)

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making a complex expression more manageable and easier to understand. When dealing with exponential expressions, like \( (x^3)^7 \), the goal is to use mathematical rules to express the expression in its simplest form. This process often involves applying specific rules of exponents to reduce the number of steps required. Simplification can help you solve problems faster and makes the expression cleaner and more intelligible.

Power of a Power Rule

The power of a power rule is a key concept when working with exponents. It simplifies expressions involving exponents raised to another exponent. The rule states that \((a^m)^n = a^{m \times n}\). This means you multiply the exponents together. Let's see how this applies to our example: \((x^3)^7\). Here, you multiply the exponents \(3\) and \(7\), resulting in \(x^{3 \times 7} = x^{21}\). This rule makes it easy to handle nested exponential expressions without having to expand them fully.

Exponentiation

Exponentiation is a fundamental operation in mathematics involving numbers, known as exponents, which represent repeated multiplication. For an expression like \(x^n\), \(n\) is the exponent telling you how many times to multiply the base, \(x\), by itself. In our solving process, we dealt with powers of \(x\) and applied the operations systematically. Understanding exponentiation helps in grasping why the power rules work and facilitates solving related problems efficiently.

Multiplying Exponents

The concept of multiplying exponents is essential in simplifying expressions with powers raised to other powers. In the context of the power of a power, you multiply the exponents directly: \( (x^m)^n = x^{m \times n} \). This direct multiplication reduces complexity and prevents the need to expand the expression step-by-step. In our example, the multiplication of \(3\) and \(7\) resulted in \(21\), giving us \(x^{21}\). Recognizing when and how to multiply exponents helps in various algebraic manipulations and problem-solving scenarios.