Problem 18 - Algebra 2 and Trigonometry

Question

Simplify each expression. In each exercise, all variables are positive. \((-2 x)^{4} \cdot\left(2 x^{3}\right)^{2}\)

Step-by-Step Solution

Answer

The expression simplifies to \(64x^{10}\).

Step 1: Apply Exponent Rule to the First Term

The expression is \[ (-2x)^4 \cdot (2x^3)^2. \]Start by simplifying each part separately. The term \[ (-2x)^4 \]denotes the exponent rule \((ab)^n = a^n \cdot b^n\). Therefore, \[ (-2x)^4 = (-2)^4 \cdot x^4. \]Calculate \((-2)^4\):\[ (-2)^4 = 16, \]which means \[ (-2x)^4 = 16x^4. \]

Step 2: Apply Exponent Rule to the Second Term

Now simplify \[ (2x^3)^2. \]Again, use the exponent rule \((ab)^n = a^n \cdot b^n\), meaning \[ (2x^3)^2 = 2^2 \cdot (x^3)^2. \]Calculate each part:\[ 2^2 = 4 \quad \text{and} \quad (x^3)^2 = x^{3 \times 2} = x^6. \]Combining these gives:\[ (2x^3)^2 = 4x^6. \]

Step 3: Multiply the Results Together

Now combine the simplified forms from Steps 1 and 2:\[ 16x^4 \cdot 4x^6. \]Multiply the constants and the variables:\[ 16 \cdot 4 = 64 \quad \text{and} \quad x^4 \cdot x^6 = x^{4+6} = x^{10}. \]Thus, \[ 16x^4 \cdot 4x^6 = 64x^{10}. \]

Key Concepts

Exponent Rules

Exponent rules are a set of laws that simplify the process of working with powers of numbers and variables. Allowing us to manage the multiplication, division, and power of a power easily, these rules are essential when tackling algebraic expressions.

• Product of Powers Rule: This rule states that when multiplying two powers with the same base, you add the exponents. Mathematically, it is represented as: \(a^m \times a^n = a^{m+n}\).
• Power of a Power Rule: When you have a power raised to another power, multiply the exponents: \((a^m)^n = a^{m\times n}\).
• Power of a Product Rule: This rule emphasizes distributing the power to each factor within the parentheses: \((ab)^n = a^n \cdot b^n\).

In the given expression, these rules help break down complex terms like \((-2x)^4\) and \((2x^3)^2\) into simpler components that can be dealt with individually.
Using the power of a product rule, \((-2)^4\) becomes \(16\) and \(x^4\) remains as \(x^4\). Similarly for \((2x^3)^2\), you get\((2^2)\) which is \(4\), and the expression \((x^3)^2\) simplifying to \(x^6\). Recognizing and applying these exponent rules properly can make algebra a lot more manageable.

Multiplying Monomials

Monomials are algebraic expressions that consist of a single term. When multiplying these, you focus on multiplying each component part—specifically the constants (numbers) and the variables (often denoted with letters representing unknowns) separately.
The expression given involves multiplying results like \(16x^4\) by \(4x^6\). Follow these steps:

• Multiply the constants: \(16\times 4 = 64\).
• Apply the product of powers rule for variables with the same base: \(x^4 \times x^6 = x^{4+6} = x^{10}\).

By handling the numbers separately from the variables, you simplify the process.
In doing so, it ensures accuracy and better clarity in your answers. Remember that the main focus is on combining like terms—those that share the same base.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators that represent a value. They are foundational in understanding algebra as they allow us to create equations and apply mathematical operations to find unknown values.
In the given task, the expression \((-2x)^4 \cdot (2x^3)^2\) is broken down using foundational algebraic principles. Each piece, or term, of the expression is simplified using exponent rules before they are multiplied together.
When you correctly organize and simplify each term, an expression like \((-2x)^4\) is transformed into \(16x^4\), and \((2x^3)^2\) becomes \(4x^6\). These parts are then multiplied to produce the final result \(64x^{10}\).
Working with algebraic expressions often involves making these tasks more approachable by applying the properties of mathematics systematically. This not only helps in understanding complex problems but also improves computational skills, ultimately broadening one's capacity in handling diverse mathematical challenges.