Question

Simplify the expression. Assume all variables are positive. \(x^9 \cdot x^2\)

Short answer

The simplified form of the expression \(x^9 \cdot x^2\) is \(x^{11}\).

Step-by-step solution

Identify the base and the exponents

Here 'x' is the base and 9 and 2 are the exponents. We have two exponential expressions with the same base being multiplied together.

Apply the product of powers rule

The product of powers rule states that for any numbers a, m, and n: \(a^m \cdot a^n = a^{m+n}\). Applying this rule to our problem, we get \(x^9 \cdot x^2 = x^{9+2}\).

Sum the exponents

Now simply add the exponents together to get \(x^{11}\).

Key concepts

Exponents Rules

Understanding exponents is essential for simplifying expressions efficiently. Exponents indicate how many times a number, known as the base, is multiplied by itself. In the expression \( x^9 \cdot x^2 \), both numbers have the same base, \(x\). The numbers \(9\) and \(2\) are the exponents.When working with exponents, there are specific rules to follow:

• Product of Powers Rule: Add the exponents when multiplying powers with the same base, e.g., \( a^m \cdot a^n = a^{m+n} \).
• Power of a Power Rule: Multiply the exponents when raising a power to another power, e.g., \( (a^m)^n = a^{m \cdot n} \).
• Power of a Product Rule: Apply the exponent to each factor inside the parenthesis, e.g., \( (ab)^n = a^n \cdot b^n \).

These rules help simplify expressions, making calculations faster and less error-prone. Remembering these rules allows students to handle a variety of algebraic expressions confidently.

Product of Powers

The Product of Powers is a fundamental rule used for simplifying expressions involving exponents, like the expression \( x^9 \cdot x^2 \). This rule states that when you multiply two powers with the same base, you can simplify by adding their exponents, which makes calculations much simpler.Here’s how it applies:

• Identify the base: In \( x^9 \cdot x^2 \), the base is \( x \).
• Add the exponents: Since both terms share the base \(x\), add the exponents \(9\) and \(2\), giving you \( x^{9+2} \).
• Simplify: This results in \( x^{11} \).

Using the Product of Powers rule effectively streamlines math problems and prepares you to handle more complex algebraic expressions with ease.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the building blocks for many higher-level mathematical problems. In our original exercise, \( x^9 \cdot x^2 \) is an example of an algebraic expression.These expressions can include:

• Constants: Numbers on their own.
• Variables: Symbols, like \( x \), that represent numbers.
• Operators: Symbols that show operations, such as addition, subtraction, multiplication, and division.

To simplify algebraic expressions, it’s important to combine like terms and use the appropriate rules for operations, such as the Product of Powers rule in our exercise.Algebraic expressions enable you to describe relationships and solve equations. They are essential in many disciplines like science, engineering, and finance, making them an important skill to master.