Question

First solve the problem, and then enter your answer on the grid provided on the answer sheet. The instructions for entering your answers follow. Given that \(x \neq 0\), find the value of \(\left(\frac{2 x^{4}+3\left(2 x^{2}\right)^{2}}{x^{4}}\right)^{2}\).

Short answer

The value of the expression is 196.

Step-by-step solution

Simplify the expression

Firstly, focus on the expression inside the fraction: \(2x^4 + 3(2x^2)^2\). Notice that \((2x^2)^2 = 4x^4\), so substitute this in to get \(2x^4 + 3 \times 4x^4\). This simplifies to \(2x^4 + 12x^4 = 14x^4\).

Simplify the fraction

Now that the numerator is \(14x^4\), simplify the fraction \(\frac{14x^4}{x^4}\). Since \(x eq 0\), you can divide both the numerator and the denominator by \(x^4\) to get \(14\).

Square the result

The simplified fraction \(\frac{14x^4}{x^4}\) gives \(14\), and you need to find the square of this result. Therefore, calculate \(14^2\), which is \(196\).

Key concepts

Algebra Simplification

Algebra simplification is akin to tidying up a math expression, ensuring we express it in the simplest and most concise form. In our problem, we started with the expression \(2x^4 + 3(2x^2)^2\). To simplify this, the first step involved recognizing that \((2x^2)^2\) can be expanded to \(4x^4\). By substituting \(4x^4\) into the expression, it transformed the equation into \(2x^4 + 3 \times 4x^4\). This step is crucial as it allows combining like terms. Like terms are terms whose variables and their exponents are the same, allowing them to be added or subtracted directly. Hence, the expression simplified further to \(2x^4 + 12x^4 = 14x^4\). This step helps in reducing complexity and is a common algebraic practice that aids in making calculations manageable.

Fraction Simplification

Fraction simplification involves reducing a fraction to its simplest form, where the numerator and the denominator have no common factors other than 1. In our initial setup, the fraction was \(\frac{14x^4}{x^4}\). Here, both the numerator and the denominator share \(x^4\) as a common factor. To simplify, we divide both the top and bottom by \(x^4\). Simplifying mathematical expressions by cancelling out common terms speeds up the solving process and eliminates unnecessary complexity, leaving us with the straightforward result of 14.Remember, fraction simplification is particularly powerful when dealing with large algebraic expressions, as it drastically reduces the amount to work with and allows focus on the most crucial parts of the problem.

Exponent Rules

Exponent rules are guidelines for simplifying expressions involving powers, and they are famously easy when you remember a few basic principles. The main properties include the product of powers rule \(a^m \times a^n = a^{m+n}\), the power of a power rule \((a^m)^n = a^{m \times n}\), and the quotient of powers rule \(\frac{a^m}{a^n} = a^{m-n}\). In our example, these principles were intertwined with simplifying \((2x^2)^2\) to \(4x^4\) by applying the power of a power rule, which states that you multiply the exponents, i.e., \((x^2)^2 = x^{2\times2} = x^4\). Once we have mastered these rules, we can tackle most problems involving exponents with confidence. They allow for the consolidation of complex expressions and make calculations more straightforward, ultimately leading to quicker solutions.