Problem 59 - Beginning and Intermediate Algebra

Question

Simplify. Assume that the variables represent nonzero integers. $$\left(g^{2 x}\right)^{4}$$

Step-by-Step Solution

Answer

The simplified expression of \(\left(g^{2x}\right)^4\) is \(g^{8x}\).

Step 1: Identifying the terms in the expression

The expression is given by \(\left(g^{2x}\right)^4\). We can identify the following components: - Base: \(g\) - Inner exponent: \(2x\) - Outer exponent: \(4\)

Step 2: Apply the power of a power rule

Now we will apply the power of a power rule, which states that \((a^m)^n = a^{mn}\). In our case, the base is \(g\), the inner exponent is \(2x\), and the outer exponent is \(4\). Plugging these into the power of a power rule, we get: \((g^{2x})^4 = g^{(2x)(4)}\)

Step 3: Simplify the expression

Now we will simplify the expression by multiplying the exponents: \(g^{(2x)(4)} = g^{8x}\) So, the simplified expression is \(g^{8x}\).

Key Concepts

Power of a Power Rule

The power of a power rule is a fundamental concept when dealing with exponents and simplifying expressions. In mathematics, this rule is used when you have an exponent that is itself raised to another exponent. It states that when you have an expression

• displayed as \( (a^m)^n \),
• you can simplify it to \( a^{m imes n} \).

This means you multiply the two exponents together.

To visualize this, consider having a base variable or number raised to one exponent, and then this entire term is again raised by another exponent. The rule makes the process straightforward by telling us to simply multiply the exponents together.

In our example, given \((g^{2x})^4\), we applied the power of a power rule:

• the inner exponent being \(2x\) and
• the outer exponent being \(4\).
• By multiplying these, we get the new exponent as \(8x\);

thus, transforming our expression to \(g^{8x}\). Utilizing this rule efficiently can simplify expressions significantly and is a crucial skill in algebra.

Exponents

Exponents are a way of representing repeated multiplication of a number by itself. They are incredibly useful in mathematics to simplify expressions and solve equations more efficiently. An exponent tells you how many times to use the number in a multiplication.

For instance, in the expression \( g^n \),

• the base is \( g \),
• and the exponent \( n \) indicates the number of times the base is used as a factor,

meaning you multiply \( g \) by itself \( n \) times.

Exponents can be whole numbers, fractions, or even variables. In the exercise given,

• the expression \( g^{2x} \)

shows how exponents can also include variables. This form lets us write very large numbers compactly and manipulate expressions in algebraic form easily. Interpreting and calculating with exponents correctly is key to solving complex mathematical problems and simplifying difficult expressions.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators (such as +, −, ×, ÷) grouped together to show the value of something. Expressions can be as simple as a single number or variable, or as complex as a series of terms connected by different operations.

The expression given in the exercise, \((g^{2x})^4\),

• involves the use of exponents,
• nested operations,
• and follows the power of a power rule to simplify it.

Breaking down an expression often involves identifying these components and understanding the relationships between them. Through simplification, we aim to reduce the complexity of these expressions, making them easier to understand and solve. This happens by applying mathematical rules and techniques.

In algebra, simplifying an expression is about removing unnecessary complexity while preserving the essential value. Understanding how to translate complex expressions into simpler ones is a core skill and builds a foundation for more advanced mathematics.