Problem 69 - Brief Applied Calculus

Question

Simplify. $$ \left[\left(x^{2}\right)^{2}\right]^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(x^8\).

Step 1: Understand the Power of a Power Rule

The expression given is \( \left(\left(x^{2}\right)^{2}\right)^{2} \). To simplify an exponent expression like this, we use the power of a power rule in exponents: \((a^m)^n = a^{m \times n}\). This means that we multiply the exponents together when raising a power to another power.

Step 2: Apply the Power of a Power Rule to the Innermost Power

First, simplify the expression \(\left(x^{2}\right)^{2}\). According to the power of a power rule, \((x^{2})^{2} = x^{2 \times 2} = x^{4}\).

Step 3: Simplify the Remaining Expression

Now, the expression is \((x^{4})^{2}\). Again, apply the power of a power rule: \(x^{4 \times 2} = x^{8}\).

Step 4: Conclude the Simplification

The simplified form of the original expression \(\left(\left(x^{2}\right)^{2}\right)^{2}\) is \(x^{8}\).

Key Concepts

Understanding the Power of a Power Rule

The power of a power rule is a fundamental concept in exponentiation. If you have an expression like \((a^m)^n\), this rule guides us to simplify it by multiplying the exponents together, becoming \(a^{m \times n}\).
This means when a quantity with an exponent is itself raised to another exponent, you don’t have to multiply the base so many times. Instead, simplify the expression by multiplying the exponents.
Using the exercise example, \(\left(x^2\right)^2\) becomes \(x^{2 \times 2} = x^4\). Once you get comfortable with this rule, you will save time and effort when dealing with complex nested exponents.

The Basics of Simplifying Exponent Expressions

Simplifying exponent expressions involves reducing them to their simplest form. This process often makes complex expressions more manageable and easier to understand.
When simplifying, the goal is to express a math problem using the least number of terms or the smallest base possible. In our example, the initial complex expression \(\left(\left(x^2\right)^2\right)^2\) gradually reduces to \(x^8\) after applying the power of a power rule twice.

• First, simplify the innermost expression: \(\left(x^2\right)^2\)
• Then reduce the overall expression: \((x^4)^2\)

Breaking it down step-by-step, as shown, helps to avoid errors and enhance your understanding of exponent rules.

Mastering Mathematical Expressions Simplification

Simplifying mathematical expressions is an essential skill, especially when dealing with algebraic problems. It involves using certain rules and properties to make expressions easier to handle.
The primary aim is to transform a complicated expression into a simpler equivalent, which typically involves fewer terms and operations.
In our exercise, simplification is achieved by recognizing and applying exponent rules to reduce \(\left(\left(x^2\right)^2\right)^2\) into \(x^8\).
This step-by-step reduction not only clarifies the problem but also highlights the power of simplification techniques, making mathematical operations more efficient and less prone to mistakes.