Problem 3 - Elementary and Intermediate Algebra

Question

Fill in the blank. a. \((3 x)^{4}=\) b. \((-5 y)(-5 y)(-5 y)=\)

Step-by-Step Solution

Answer

a. \(81 x^4\) b. \(-125 y^3\)

Step 1: Expand the Power

For part a, Start by expanding the expression for \((3x)^4\). This means raising both 3 and \(x\) to the power of 4. So the expression becomes: \((3^4)(x^4)\).

Step 2: Calculate the Numerical Power

Next, calculate \(3^4\). This is calculated as:\[3^4 = 3 \times 3 \times 3 \times 3 = 81\] Thus, \((3x)^4 = 81x^4\).

Step 3: Simplify the Multiplication

For part b, write out the expression: \((-5y)(-5y)(-5y)\).Use the associative property to first calculate \((-5y)(-5y) = 25y^2\), because\((-5)^2 = 25\) and \(y \times y = y^2\).

Step 4: Final Multiplication to Simplify

Multiply the result from Step 3 by \(-5y\).\((25y^2)(-5y) = -125y^3\), because \(25 \times -5 = -125\) and \(y^2 \times y = y^3\).

Key Concepts

Polynomial Expressions

Polynomial expressions are algebraic expressions that consist of variables and coefficients, structured in terms of powers raised to whole numbers. They often include a sum of terms, where each term is a product of a constant (coefficient) and one or more variables raised to a power.

Some common examples of polynomial expressions are:

• \(3x^2 + 2x + 1\)
• \(4x^3 - 7x + 5\)
• \(x^4 + 3x^3 + 3x^2 + x + 1\)

When dealing with polynomial expressions, they are often simplified or manipulated in algebra to solve equations or simplify forms. Key to working with them is understanding how to apply operations, like addition, subtraction, multiplication, and division, alongside using laws of exponents to combine like terms or expand expressions.

Power of a Product

The power of a product rule is a vital concept in algebra that simplifies expressions where a product is raised to an exponent. When you have a product consisting of two or more factors raised to a power, each factor is raised to the power independently. This is expressed as:

• \((ab)^n = a^n \cdot b^n\)

For instance, in

• \((3x)^4 = (3^4)(x^4)\)

Each factor in the product, 3 and \(x\), is raised to the power of 4.

This rule helps in simplifying expressions before further operations are performed, ensuring a more manageable form for both computation and simplification. It underscores the separable nature of a product when raised to an exponent, which is invaluable in broader polynomial manipulations.

Properties of Exponents

Understanding the properties of exponents is crucial for working with algebraic expressions efficiently. These properties provide rules for how to handle exponents when they appear in mathematical expressions. Here are some of the key properties:

• Product of Powers Property: When multiplying two terms with the same base, you add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• Power of a Power Property: When raising a power to another power, you multiply the exponents: \((a^m)^n = a^{m \cdot n}\).
• Power of a Product Property: As explored earlier, apply the exponent to each factor in the product: \((ab)^n = a^n \cdot b^n\).
• Negative Exponent Property: An exponent of zero means the result is 1: \(a^0 = 1\) (assuming \(a eq 0\)), and a negative exponent implies a reciprocal: \(a^{-n} = \frac{1}{a^n}\).

These properties allow for the streamlining of expressions and are particularly useful in solving equations, simplifying complex algebraic problems, and performing polynomial long division. They are fundamental tools in ensuring that one can manipulate and evaluate expressions thoroughly and accurately.