Question

Simplify. $$ (7 x)^{2} $$

Short answer

49x^{2}.

Step-by-step solution

Identify the terms inside the parentheses

The terms inside the parentheses are 7 and x.

Apply the exponent to each term within the parentheses

The problem simplifies to (7)^{2} * (x)^{2}.

Simplify further

Now, (7)^{2} equals 49 and (x)^{2} equals x^{2}. So the final answer is 49x^{2}.

Key concepts

Exponent Rules

When simplifying algebraic expressions involving exponents, understanding the exponent rules is essential. One of the fundamental rules is the 'Power of a Product' rule, which states that when you raise an entire product to a power, you can apply the exponent to each factor separately. For example, in the expression \( (7x)^2 \), the 'Power of a Product' rule allows us to simplify this to \( 7^2 * x^2 \). Each factor inside the parentheses is raised to the power of 2 independently. Remember to treat each component with care, squaring the numerical part and the variable part separately to avoid common errors.

Order of Operations

Simplifying expressions correctly often relies on following the order of operations, a fundamental concept in algebra. This convention dictates the sequence in which mathematical operations should be performed. The order of operations is usually remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). By adhering to this order, you ensure that your calculation steps are consistent with mathematical standards. So, when you approach an expression like \( (7x)^2 \), you first observe the parentheses and then apply the exponent, as described in the 'Power of a Product' rule. This ensures that you work out the problem systematically and correctly.

Exponential Expressions

Exponential expressions frequently appear in algebra, and it's crucial to know how to handle them. An exponential expression has a base and an exponent, as seen in \( x^2 \) where \( x \) is the base, and 2 is the exponent. The exponent tells us how many times to multiply the base by itself. Therefore, \( x^2 \) means \( x \) multiplied by \( x \) once. Similarly, \( 7^2 \) equates to 7 multiplied by itself, resulting in 49. These expressions can be constants, like \( 7^2 \), or variables, like \( x^2 \) and understanding how to work with both in simplifying algebraic expressions is vital for algebraic proficiency.