Problem 40 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The expression \((8x)^2\) simplifies to \(64x^2\).

Step 1: Understand the Expression

The expression \((8x)^2\) means that we need to square the term \(8x\). Squaring a term is equivalent to multiplying it by itself.

Step 2: Apply the Square Operation

To square the expression \((8x)^2\), multiply \(8x\) by itself: \((8x) \times (8x)\).

Step 3: Square the Coefficient

Multiply the numerical coefficients: \(8 \times 8 = 64\). This gives us part of the squared term: \(64x^2\).

Step 4: Square the Variable

Multiply the variable parts by applying the exponent rule: \(x \times x = x^2\).

Step 5: Combine the Results

Combine the squared coefficient and variable to get the simplification: \(64x^2\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number or expression to a particular power. In the simplest terms, when we see a number or variable raised to a power—like in the expression \[ (8x)^2 \] —we are using exponentiation. The power, denoted by the small number above and to the right of the base, tells us how many times to use the base in a multiplication.
For example, squaring, which is an exponent of 2, means multiplying the base by itself. So, in our example, we took the term \[ 8x \] and multiplied it by itself: \[ (8x) \times (8x) \].
This operation is important because it links to several other mathematical concepts, like powers of variables in polynomial expressions and the laws of exponents, which include rules like \[ (a^m)^n = a^{m\times n} \]. Learning how to handle exponentiation is crucial for simplifying expressions correctly.

Polynomial Expression

A polynomial expression is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The expression \[ 64x^2 \] is a simple example of a polynomial. It is a monomial because it consists of only one term.
Polynomial expressions can be more complex, with multiple terms, each having variables raised to various powers.

• Coefficients are the numbers that multiply the variables.
• The degree of a polynomial is the highest power of any variable within the expression.

In the simplification of \[ (8x)^2 \], we started with an expression with one term and, through the operation of squaring, maintained a monomial expression but with altered coefficients and variables' exponents. Understanding polynomials helps in algebra because they form the basis of many equations and functions.

Mathematical Operations

Mathematical operations refer to techniques used to manipulate and simplify expressions, which include addition, subtraction, multiplication, division, and exponentiation. In algebra, these operations are vital for simplifying and solving problems.
In the example \[ (8x)^2 \], we used multiplication and exponentiation as part of the simplification process. Here's how these operations unfolded:

• Multiplying the coefficient: Multiply 8 by itself to get 64.
• Multiplying the variable: Use the exponentiation rule \[ x \times x = x^2 \]. This results in the final expression: \[ 64x^2 \].

By understanding these operations, we can simplify and resolve algebraic expressions more effectively. Each operation has specific rules and properties that, when understood, make problem-solving much easier. They work hand-in-hand, allowing us to break down complex problems into simpler, more manageable parts.