Problem 46 - College Algebra

Question

Simplify each exponential expression in Exercises 23–64. $$\left(11 x^{5}\right)\left(9 x^{12}\right)$$

Step-by-Step Solution

Answer

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

Step 1: Identify the bases that are the same

In this equation we see that \(x\) is the base that is the same in both expressions that are multiplied.

Step 2: Multiplication of the coefficients

Multiply the coefficients 11 and 9 together to get 99. This will result in \(99 x^{5}x^{12}\).

Step 3: Addition of exponents

We apply the addition rule for exponents: when multiplying two expressions with equal bases, we can add the exponents together. So, we add the exponents 5 and 12 together to get 17. This simplifies the expression to \(99x^{17}\).

Key Concepts

Exponential Rules

When dealing with exponential expressions, it's important to understand the fundamental rules that govern their behavior. One of the most important things to remember is that when you are multiplying numbers with the same base, you keep the base the same and add the exponents. For instance, in the expression \(a^m \cdot a^n\), the result would be \(a^{m+n}\).

Another important rule to keep in mind is the power of a power rule: \( (a^m)^n = a^{mn} \). This means if an exponential expression is raised to another power, you multiply the exponents. Meanwhile, the power of a product rule states that \( (ab)^n = a^n \cdot b^n \), which means that you can distribute the exponent to each factor in the product.

For division, there's a rule too: \( \frac{a^m}{a^n} = a^{m-n} \(a\eq0\) \), where you subtract the exponent in the denominator from the exponent in the numerator, keeping the base unchanged if they are the same.

Multiplication of Coefficients

Coefficients are the numerical parts of algebraic terms, which are often found in front of variables. When you multiply two terms that have coefficients, you simply multiply the coefficients together, independent of the variables. For example, if we have the expression \(11x^5\) being multiplied by \(9x^{12}\), then we multiply the coefficients 11 and 9 to get 99.

It's essential to treat coefficients separately from the variables they are associated with. In our example, multiplying the coefficients results in an expression like \(99x^5x^{12}\). This step is valuable because it simplifies the expression and prepares us for further simplification using the properties of exponents.

Addition of Exponents

When adding exponents, the base of the terms must be the same. This principle is key in simplifying exponential expressions. In our previously mentioned expression \(99x^5x^{12}\), since both terms have the same base (\

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. In the example of simplifying \(11x^5\) times \(9x^{12}\), we encountered such an algebraic expression. In this case, we identified the variables and coefficients and applied the appropriate rules to simplify the expression.

Learning to work with these expressions requires practice and a strong grasp of the arithmetic operations and properties of exponents. One tip for tackling algebraic expressions is to always perform operations in a systematic order: deal with coefficients, manage similar bases by adding or subtracting exponents, and finally, carry out any additional necessary operations to fully simplify the expression.