Question

Write each expression without parentheses. Assume all variables are positive. $$ \left(3 x^{2}\right)^{2 n} $$

Short answer

Question: Rewrite the given expression without parentheses: \((3x^2)^{2n}\) Answer: The rewritten expression without parentheses is: $$3^{2n} x^{4n}$$.

Step-by-step solution

Apply the Exponent Rule for Products

We are given the expression: \((3x^2)^{2n}\). The first step is to apply the exponent rule for products: \((ab)^n = a^n b^n\). In this case, \(a = 3\), \(b = x^2\), and \(n = 2n\). Applying this rule to the expression, we get: $$(3x^2)^{2n} = 3^{2n} (x^2)^{2n}$$

Apply the Exponent Rule for Exponents of Exponents

Now, let's apply the exponent rule for exponents of exponents: \((a^{n})^m = a^{nm}\). In this case, for the second term, \(a = x, n = 2\), and \(m = 2n\). Applying this rule, we get: $$3^{2n} (x^2)^{2n} = 3^{2n} x^{(2)(2n)}$$

Simplify the Expression

Finally, we will simplify the expression by multiplying the exponents in the second term: $$3^{2n} x^{(2)(2n)} = 3^{2n} x^{4n}$$ The expression without parentheses is: $$3^{2n} x^{4n}$$

Key concepts

Exponent Rules

When working with algebraic expressions, exponent rules are crucial for simplifying terms with exponents. One key rule is the product of powers rule, which states that when multiplying like bases, you add their exponents: \((a^m)(a^n) = a^{m+n}\). This rule helps to combine expressions with the same base.
Another useful rule is the power of a power rule. This states that when raising a power to another power, you multiply the exponents: \((a^m)^n = a^{mn}\). For example, in the expression \((3x^2)^{2n}\), the base \(3x^2\) is raised to the \(2n\) power.
It’s essential to apply these rules correctly, as they determine how you simplify or rewrite expressions without parentheses, just like in our example. Each component must be raised independently, adhering to the relevant exponent rule.

Simplifying Expressions

Simplifying expressions involves reducing them down to their simplest form while still keeping their value unchanged. It requires understanding the structures of expressions, such as terms, coefficients, and exponents.
To simplify \( (3x^2)^{2n} \), we first separate the components using exponent rules. This involves expanding the expression by handling the numbers and variables separately. For the constant, \(3^{2n}\) follows directly from applying the power of a product rule. Then, within \(x^2^{2n}\), we use exponent rules to simplify further to \(x^{4n}\).

• First, handle constants and variables on their own.
• Use exponent rules consistently.
• Keep multiplying exponents where necessary.

Simplifying step-by-step ensures no part of the expression is overlooked and helps in understanding transformations thoroughly.

Variables in Algebra

In algebra, variables like \(x\) serve as placeholders for unknown values or numbers. They are fundamental in creating expressions and equations that model real-world scenarios or abstract problems.
With expressions such as \( (3x^2)^{2n} \), variables show how changes affect the overall value. The exponentiation of variables can imply complex relationships between quantities, as seen with \(x^{4n}\), meaning \(x\) is raised to a higher power, influenced by \(n\).
Understanding variables involves:

• Recognizing their role as symbols representing numbers.
• Comprehending how operations like addition, multiplication, and exponentiation impact them.
• Using variables to express general rules or patterns.

They are essential tools in algebra, allowing for flexible and dynamic expressions that can be adapted when specific values become known.