Problem 29 - Algebra and Trigonometry

Question

\(29-46\) Simplify each expression. $$ x^{8} x^{2} $$

Step-by-Step Solution

Answer

The expression simplifies to \( x^{10} \).

Step 1: Understand the Problem

The expression given is a multiplication of two powers with the same base, which is \( x \). We're asked to simplify this expression.

Step 2: Recall the Exponent Rule

When multiplying powers with the same base, we use the rule \( a^m \times a^n = a^{m+n} \). This rule allows us to add the exponents together.

Step 3: Apply the Exponent Rule

Using the rule from Step 2, we can simplify \( x^8 \times x^2 \) by adding the exponents: \( 8 + 2 = 10 \).

Step 4: Write the Simplified Expression

The expression simplifies to \( x^{10} \).

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making mathematical expressions easier to read and solve. When you simplify, you're essentially cutting down on complexity. Take the example expression from the exercise: \[ x^8 \times x^2. \]The aim is to express this in the simplest form possible. To achieve this, we apply mathematical rules that reduce the expression to fewer terms. In our case, we want to combine the exponents of the same base. We should also pay close attention to other simplification opportunities, such as combining like terms or canceling out factors, if applicable. The simplified result is cleaner and easier to work with.

Multiplication of Powers

The multiplication of powers with the same base is a key operation in algebra. When you multiply expressions that have the same base, you're essentially combining them into a single power. Let's look at our example: \[ x^8 \times x^2. \]Both terms share the base 'x'. According to the rules of exponents, when you multiply two powers that share the same base, you keep the base and add the exponents. So, you do not multiply the base values; instead, you operate on the exponents. This helps in simplifying the expression efficiently, reducing multiple terms into one. In our example, you simply add 8 and 2 to get \[ x^{10}. \] This process makes the expression concise and easy to handle in further calculations.

Exponent Rules

Exponent rules are powerful guidelines that help us maneuver through expressions involving powers. These rules give us the mathematical power (pun intended) to simplify expressions efficiently and accurately. One crucial rule to remember is:

• When multiplying powers with the same base, add the exponents: \( a^m \times a^n = a^{m+n} \).

With this rule, you can handle even the toughest expressions. Such rules ensure consistency and reliability in mathematical operations. They also allow for quicker computation of seemingly complex expressions. In the exercise, using the rule where we add the exponents for \( x^8 \times x^2 \), we got \( x^{10} \). Being familiar with these and other rules like the power of a power or zero exponent rule will make dealings with exponential expressions a breeze.