Problem 32 - Algebra and Trigonometry

Question

Simplify each exponential expression. $$ \left(x^{11}\right)^{5} $$

Step-by-Step Solution

Answer

The simplified form of the expression \((x^{11})^{5}\) is \(x^{55}\)

Step 1: Identify the Components of the Expression

In the expression \((x^{11})^{5}\), \(x\) is the base, 11 is the first exponent and 5 is the second exponent.

Step 2: Apply the Power of a Power Rule

The Rule of Exponents for Power of a Power tells us that when we have one exponent raised to another power, we multiply the two exponents together. So \((x^{11})^{5}\) can be simplified as \(x^{11*5}\).

Step 3: Simplify the Expression

After multiplying the two exponents together, we get \(x^{55}\).

Key Concepts

Power of a Power Rule

When you encounter expressions like \( (x^{11})^{5} \), you're seeing the 'power of a power rule' at play. This rule is a foundational concept in algebra that simplifies expressions where an exponent is raised to another exponent. You can think of it like a shortcut for repeated multiplication.

For example, if you want to expand \( (x^{2})^{3} \), using the power of a power rule, you would multiply the exponents (2 and 3 in this case) rather than writing out \( x^{2} \times x^{2} \times x^{2} \). Applying this rule, \( (x^{2})^{3} \) simplifies to \( x^{2*3} = x^{6} \).

Back to our original exercise, \( (x^{11})^{5} \) simplifies by multiplying the exponents 11 and 5 to get \( x^{11*5} = x^{55} \), which gives you the final simplified form.

Exponents Multiplication

Learning about exponents, you'll soon discover that there are rules to make calculations easier, especially with multiplication. Exponents multiplication, specifically, deals with scenarios where you have the same base being raised to different powers and need to multiply them. The key is understanding that when the bases remain constant and only the exponents change, you simply add the exponents to simplify the expression.

Consider \( x^{a} \times x^{b} \); it simplifies to \( x^{a+b} \). However, there's a distinct difference in the application from the 'power of a power rule.' While the latter multiplies the exponents, exponents multiplication involves addition.

This distinction helps avoid confusion and ensures correct simplification in various algebraic expressions.

Algebraic Simplification

Algebraic simplification is the process of making expressions more concise and manageable without altering their value. It's like decluttering a room: you're organizing and getting rid of unnecessary complexity to reveal a clearer space—or in this case, a clearer mathematical expression.

Simplifying an expression often involves applying a variety of exponent rules, like the power of a power rule and exponents multiplication discussed earlier. It could also involve combining like terms, factoring polynomials, or using the distributive property to condense an expression to its simplest form.

Understanding these rules and when to apply them reduces mistakes and makes solving algebraic problems more efficient. It's essential to practice these concepts repeatedly to recognize patterns and become proficient at simplifying complex algebraic expressions quickly and correctly.