Problem 79 - Algebra A Combined Function

Question

Simplify each expression. $$ \left(2 x^{3}\right)\left(-8 x^{4}\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \(-16x^7\).

Step 1: Multiply the Coefficients

To begin simplifying the expression \((2x^3)(-8x^4)\), first multiply the numerical coefficients. The coefficients are 2 and -8. Multiply these together: \[2 \times -8 = -16\]

Step 2: Apply the Product of Powers Rule

Next, we apply the product of powers rule to the variable terms. The rule states that when multiplying like bases, you add the exponents. Here, we have: \[x^3 \cdot x^4 = x^{3+4} = x^7\]

Step 3: Combine Results

Combine the results from the previous steps. The multiplication of the coefficients gave us \(-16\) and the multiplication of the variable terms gave us \(x^7\). Thus, the simplified expression is:\[-16x^7\]

Key Concepts

Coefficients

In algebraic expressions, coefficients are the numerical factors in terms that include a variable. For instance, in the expression \(2x^3\), the number 2 is the coefficient. Coefficients represent the number of times the variable is multiplied. Whenever simplifying expressions, you simply multiply the coefficients across terms. The process is straightforward:

• Identify the coefficients in each term. In our example, they are 2 and -8.
• Multiply these coefficients together, as follows: \(2 \times -8 = -16\).

This multiplication step means that any algebraic manipulation keeps both the relative size and direction (positive or negative) of the expression intact.

Product of Powers Rule

The product of powers rule is a fundamental principle in algebra that applies when multiplying two powers with the same base. This rule states that to multiply these powers, you add their exponents together. In more formal terms, for any base \(a\), the rule is defined as:

• \(a^m \cdot a^n = a^{m+n}\)

Consider the expression \(x^3 \cdot x^4\). Applying the product of powers rule, you add the exponents 3 and 4:

• \(x^{3+4} = x^7\)

This rule helps simplify expressions by reducing multiple terms into a single term, making computations easier and clearer.

Exponents

Exponents, also known as powers, are a way of expressing repeated multiplication of a number or variable by itself. In \(x^3\), the number 3 is the exponent and indicates that the base, \(x\), is multiplied by itself three times: \(x \times x \times x\).In algebraic operations involving exponents:

• When multiplying like bases, add the exponents: \(x^a \cdot x^b = x^{a+b}\).
• When dividing like bases, subtract the exponents: \(x^a / x^b = x^{a-b}\).

Understanding exponents is crucial, as they vastly simplify expressions that otherwise involve lengthy multiplications.

Simplifying Expressions

Simplifying expressions means to reduce them to their most concise form without changing their value. The goal is to make expressions easier to handle and understand.The process often involves several steps:

• First, simplify any coefficients by multiplication or division as required.
• Then, apply rules of exponents such as the product of powers for like bases.
• Combine all elements into a single, simplified expression.

For instance, in simplifying \((2x^3)(-8x^4)\):

• Multiply the coefficients: \(2 \times -8 = -16\).
• Apply the product of powers rule to \(x^3 \cdot x^4\): \(x^{3+4} = x^7\).
• Combine results: \(-16x^7\).

Simplification not only makes expressions more elegant but also facilitates easier computation in algebraic problem-solving.