Problem 46 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Properties of Exponents

The product-of-powers property of exponents states that when multiplying like bases, the exponents are added. This will be applied to the given expression, which is \( (11 x^{5})(9 x^{12}) \)

Step 2: Apply the Property of Exponents

Apply the product-of-powers property to the expression. This means adding the exponents while multiplying the coefficients 11 and 9. So, the simplified expression becomes \(11 * 9 * x^{5+12}\)

Step 3: Simplify the Expression

Further simplify the expression by completing the multiplication and adding the exponents. The final simplified expression becomes \(99 * x^{17}\)

Key Concepts

Properties of Exponents

Understanding the properties of exponents is essential for simplifying complex algebraic expressions. Exponents, also known as powers, indicate how many times a base number is to be multiplied by itself. When dealing with exponential expressions, certain rules or properties allow us to perform operations more easily.

Here are some basic properties you should know:

• The Product of Powers property says that when you multiply two powers with the same base, you simply add the exponents.
• The Quotient of Powers property tells us that when we divide two powers with the same base, we subtract the exponents.
• The Power of a Power property means that when you raise an exponent to another exponent, you multiply the exponents.
• The Zero Exponent Rule states that any base raised to the power of zero is equal to one.
• The Negative Exponent Rule implies that a negative exponent indicates the reciprocal of that base raised to the opposite positive exponent.

When simplifying exponential expressions using these properties, it's important to apply each property correctly to avoid mistakes and ensure the expression is as simple as possible.

Product-of-Powers Property

The product-of-powers property is a specific rule within the properties of exponents that comes into play when multiplying expressions with the same base. According to this property, given two exponents with the same base, you can simply add the exponents while keeping the base unchanged.

For example, consider the expression \( x^a \times x^b \). Instead of multiplying \( x \) by itself \( a \) times, and then \( b \) times separately, we can use the product-of-powers property to write \( x^{a+b} \).

In practice, this property makes it easier to deal with multiplication involving exponents. To apply this rule effectively, always make sure that the bases you're multiplying are the same and that you are only adding exponents, not the bases themselves. Keep in mind that coefficients in front of the bases are multiplied together normally, as shown in the example \((11 x^{5})(9 x^{12}) = 99 x^{17}\), following this product-of-powers property.

Exponential Notation

Exponential notation is a mathematical shorthand used to express repeated multiplication of a number by itself. This notation consists of two parts: the base and the exponent. The base is the number being multiplied, and the exponent tells us how many times the base is used as a factor in the multiplication.

In the expression \( x^n \), \( x \) is the base and \( n \) is the exponent. The exponent is placed as a superscript to the right of the base. It's important to understand that \( x^n \) is not the same as \( x \times n \); instead, \( x^n \) means \( x \times x \times ... \times x \) (\( n \) times). For instance, \( 3^4 \) is \( 3 \times 3 \times 3 \times 3 \), which equals 81.

When simplified, this notation helps us to write and work with very large or very small numbers efficiently. The use of exponential notation is widespread in various fields of science, economics, and engineering because it allows for the easy comprehension and manipulation of numbers that are otherwise too big or too small to be dealt with straightforwardly.