Problem 66 - 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. $$ \left(x^{n} t^{2 m}\right)^{4} $$

Step-by-Step Solution

Answer

Question: Simplify the following expression: \(\left(x^{n} t^{2 m}\right)^{4}\) Answer: \(x^{4n} t^{8m}\)

Step 1: Understand the power rules

In this problem, we will use the power rule that states \((a^m b^n)^p = a^{m*p} b^{n*p}\).

Step 2: Apply the power rule to the given expression

Now, we can apply this rule to the expression \(\left(x^{n} t^{2 m}\right)^{4}\): $$ \left(x^{n} t^{2 m}\right)^{4} = x^{n*4} t^{(2 m)*4} $$

Step 3: Simplify the expression

Now we can simplify the expression by multiplying the exponents: $$ x^{n*4} t^{(2 m)*4} = x^{4n} t^{8m} $$ So the simplified expression is: $$ \left(x^{n} t^{2 m}\right)^{4} = x^{4n} t^{8m} $$

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving numbers or expressions called bases and exponents. It is written in the form \(a^n\), where \(a\) is the base and \(n\) is the exponent. Exponentiation represents repeated multiplication of the base by itself. For example, \(3^4\) means \(3 \times 3 \times 3 \times 3\). It simplifies expressions and helps solve complex algebraic equations.
When dealing with expressions with exponents, remember:

• A positive exponent indicates how many times to multiply the base by itself.
• A zero exponent indicates the result is always 1, as any number raised to the power of zero is 1.
• A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent, such as \(a^{-n} = \frac{1}{a^n}\).

Understanding exponentiation allows you to work with power laws, which are essential for simplifying complex algebraic expressions.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators (like addition and multiplication) that represents a mathematical relationship. For example, \(3x + 2y - 5\) is an algebraic expression where \(x\) and \(y\) are variables.
Key elements of algebraic expressions include:

• Coefficients: Numbers that multiply the variables, such as 3 in \(3x\).
• Variables: Symbols, usually letters, representing unknown values or quantities.
• Constants: Fixed values within the expression, like \(-5\) in \(3x + 2y - 5\).
• Operators: Indicate mathematical operations; for example, '+', '-', '*' and '/'

Algebraic expressions are the foundation of algebra and are used to form equations and functions. Simplifying them involves reducing expressions to their simplest form, often using rules like distribution and combining like terms.
Grasping the components of algebraic expressions is crucial in effectively working through algebra problems.

Simplification in Algebra

Simplification in algebra involves reducing expressions to their most concise form without changing their value. This process utilizes several key principles and rules, ensuring clear and efficient problem-solving.
To simplify algebraic expressions, it is often necessary to:

• Apply the laws of exponents, like the power rule: \((a^m b^n)^p = a^{m*p} b^{n*p}\).
• Multiply coefficients and add exponents when bases are the same, such as in \(x^3 \cdot x^2 = x^{3+2} = x^5\).
• Distribute multiplication over addition, \(a(b+c) = ab + ac\).
• Combine like terms, which are terms that contain the same variables raised to the same exponent, such as \(2x + 3x = 5x\).

Simplification is essential because it makes expressions and equations more manageable and easier to understand. It is fundamental in solving algebra problems, enabling you to find solutions in a systematic and organized manner. Understanding and mastering these simplification techniques will help you handle more advanced mathematics with confidence.