Problem 69 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ (2 a b)^{t} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((2ab)^t\). Answer: $$2^t a^t b^t$$

Step 1: Distribute the exponent to each term inside the parentheses

We will distribute the exponent \(t\) according to the first power rule: \((ab)^n = a^n b^n\). In this case, we have \((2ab)^t\), and we will apply this rule to simplify the expression. \((2ab)^t = 2^t a^t b^t\) Step 2: Final simplified expression

Step 2: Write the final simplified expression

After distributing the exponent, our final simplified expression is: $$2^t a^t b^t$$

Key Concepts

Power Rules for Exponents

One essential aspect of algebra is understanding how to deal with exponents, and the power rules for exponents are critical for simplifying exponential expressions. When an expression with a power is raised to another power, we use the power of a power rule, which states that \( (a^m)^n = a^{m \cdot n} \). This means you multiply the exponents. Similarly, when you have a product raised to a power, like \( (ab)^n \), you apply the power to both components, resulting in \( a^n \cdot b^n \).

Applying this to the exercise, \( (2 a b)^{t} \) simplifies to \( 2^t a^t b^t \), by multiplying the base 2, a, and b, each by the exponent \( t \). Remember, these rules help ensure that the process of simplification stays structured and easy to follow.

Distributive Property of Exponents

The distributive property of exponents refers to how we deal with exponents when they are distributed over a multiplication or division within an expression. It falls under two main scenarios:

• When a product like \( ab \) is raised to a power \( t \) as in our exercise \( (2ab)^t \), we can distribute the exponent to each factor resulting in \( 2^t \cdot a^t \cdot b^t \).
• When dividing two terms with the same base but different exponents, you subtract the exponents, which is not the focus of our current exercise but still a useful rule to know.

In the case of our exercise, simplifying \( (2ab)^t \) utilizes the distributive property by applying the exponent \( t \) to each base within the parentheses. This property illustrates how exponents interact with basic arithmetic operations.

Exponentiation

Exponentiation is the mathematical operation involving two numbers, the base \( a \) and the exponent or power \( n \) in the expression \( a^n \). The exponent expresses the number of times the base is multiplied by itself. For example, \( 3^4 \) means \( 3 \times 3 \times 3 \times 3 \).

In our exercise, when we see \( (2ab)^t \), we understand that exponentiation tells us to multiply \( 2ab \) by itself \( t \) times. However, simplifying this kind of expression usually does not require performing all the multiplications; instead, we apply rules of exponents to make the process more efficient. Therefore, understanding exponentiation is not just about calculating powers, but also about knowing how to use exponent rules to simplify complex expressions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. Variables represent unknown values and can be any letter. In algebra, you often see expressions like \( 2a \) where \( 2 \) is the coefficient and \( a \) is the variable. When simplifying algebraic expressions, you apply laws of arithmetic and rules of exponents.

The expression \( (2ab)^t \) from our exercise showcases a simple algebraic expression raised to a power. To simplify this expression, you must recognize the interaction between coefficients (in this case, \( 2 \)), variables (\( a \) and \( b \)), and exponents (\( t \)). By understanding how these components work together, you can apply the appropriate algebra rules to simplify an expression effectively and reach the final simplified form, proving the usability of algebra in simplifying complex numerical relationships.