Problem 70 - Elementary and Intermediate Algebra

Question

Use the power of a product rule for exponents to simplify each expression. $$ (4 t)^{4} $$

Step-by-Step Solution

Answer

The simplified expression is \(256 t^4\).

Step 1: Understand the Power of a Product Rule

The power of a product rule states that for any two numbers \(a\) and \(b\), and any positive integer \(n\), \((a \, b)^n = a^n \, b^n\). This means when you raise a product to a power, you can apply the exponent to each factor separately.

Step 2: Apply the Rule to the Expression

Using the power of a product rule, let's apply the exponent 4 to each factor inside the parentheses separately in \((4t)^4\). This means: \((4 \cdot t)^4 = 4^4 \cdot t^4\).

Step 3: Calculate the Power of 4

Now calculate \(4^4\). This means multiplying 4 by itself 4 times: \(4 \times 4 \times 4 \times 4 = 256\).

Step 4: Write the Final Expression

Substitute the value of \(4^4\) back into the expression to obtain the simplified form: \(256 \cdot t^4\). Thus, the simplified expression of \((4t)^4\) is \(256 t^4\).

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation that involves raising a number to the power of another number. It answers the question: How many times do we multiply the base by itself? For example, in the expression \( 4^4 \), 4 is the base and 4 is the exponent, meaning we multiply 4 by itself four times as in \( 4 \times 4 \times 4 \times 4 \). The result is 256.
Exponentiation is crucial in algebra, providing a way to express large numbers easily. It is used in various fields such as finance for calculating compound interest, in physics for exponential growth or decay, and in computer science for algorithms dealing with large data sets.
Understanding the rules of exponentiation, like the power of a product rule, allows you to simplify expressions efficiently. This also sets a strong foundation for tackling more complex algebraic operations.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra. It involves reducing a mathematical expression to its simplest form. The goal is to make the expression as easy to work with as possible, facilitating further operations or understanding.
When simplifying, you might need to:

• Combine like terms, such as \( x \) and \( 2x \), to form \( 3x \).
• Apply laws of exponents, such as the power of a product rule, to manage complex expressions.
• Cancel out terms where possible, though this isn't applicable to exponentiation directly.

For the expression \((4t)^4\), using the power of a product rule allowed us to break it into \(4^4 \cdot t^4\), making calculations more straightforward. Mastering these steps streamlines problem solving in algebra.

Algebra

Algebra forms the backbone of advanced mathematics, providing tools to solve a variety of problems. In algebra, expressions can contain constants, variables, and operations. It involves working with unknowns represented by symbols and finding their values.
In this context, the expression \((4t)^4\) is algebraic because it consists of a numerical coefficient (4) and a variable (\(t\)). We apply rules, like the power of a product, to manipulate the expression. This is part of learning how to rearrange and simplify equations, which prepares you for solving equations.
Algebra is used in fields ranging from engineering to economics. Understanding its principles, like simplification and exponentiation, is essential for progressing to complex topics such as calculus and differential equations. By honing algebraic skills, one lays a solid foundation for academic and professional success.