Problem 46 - Algebra and Trigonometry

Question

Simplify each exponential expression. $$ \left(11 x^{5}\right)\left(9 x^{12}\right) $$

Step-by-Step Solution

Answer

The simplified form of the exponential expression \((11x^{5})\cdot(9x^{12})\) is \(99x^{17}\).

Step 1: Identify exponential terms

The expression consists of two exponential terms \((11x^5)\) and \((9x^{12})\). Here, \(x\) is the base, \(5\) and \(12\) are the exponents for the first and second term, respectively. Keep in mind that each exponential term also includes a coefficient: \(11\) for the first term and \(9\) for the second.

Step 2: Multiply the coefficients and add the exponents

According to the rules of exponents, when you have the same base, you can add the exponents together during multiplication. So, multiply \(11\) and \(9\), multiply \(x^5\) and \(x^{12}\), and combine the results. This makes the calculation straightforward: \(11 \cdot 9 \cdot x^{5+12} = 99x^{17}\).

Step 3: Present simplified exponential expression

The exponential expression \(99x^{17}\) is the simplified result.

Key Concepts

Exponents

Exponents are shorthand for repeated multiplication. In an expression like \( x^n \), \( x \) is the base, and \( n \) is the exponent, indicating that \( x \) is to be multiplied by itself \( n \) times. For example, \( x^3 \) is \( x \times x \times x \). Exponents can dramatically change the value of an expression, especially as the exponent increases.

Understanding the concept of an exponent is crucial in algebra because it helps clarify how numbers and variables can grow or decrease rapidly with each power. The impact of exponents on calculations and their related algebraic rules allow for easier manipulation and simplification of complex expressions, providing a more efficient path to solutions.

Exponential Terms

An exponential term involves a base raised to a power, like \( a^b \), but it can also include a coefficient, as seen in \( 11x^5 \). The coefficient is the number you multiply by the exponential part of the term. It's important to differentiate between the coefficient and the base since they follow different rules during simplification.

When working with exponential expressions, it is essential to identify all parts correctly to apply the correct operations. For example, in the expression \( 11x^5 \), \( 11 \) is the coefficient, and \( 5 \) is the exponent of the base \( x \). Incorrectly identifying these components can lead to errors in simplification and further calculations.

Rules of Exponents

The rules of exponents are guidelines that describe how to handle exponential expressions when multiplying, dividing, or raising powers to other powers. Key rules include:

• Product of Powers: When multiplying with the same base, you add the exponents. For example, \( x^a \times x^b = x^{a+b} \).
• Quotient of Powers: When dividing with the same base, you subtract the exponents. For instance, \( x^a \div x^b = x^{a-b} \).
• Power of a Power: When you have a power raised to another power, you multiply the exponents. For example, \( (x^a)^b = x^{a\times b} \).

Applying these rules, we can simplify complex expressions accurately. During the process of simplifying the exercise given, \((11 x^{5})\times(9 x^{12})\), we multiplied coefficients and added exponents of like bases, resulting in the simplified exponential expression of \(99x^{17}\). This adheres to the product of powers rule and is a fundamental aspect of algebra.