Problem 75 - 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: Apply the Power of a Power Rule

The expression given is \( \left[\left(x^{2}\right)^{2}\right]^{2} \). The power of a power rule states that \( \left( a^{m} \right)^{n} = a^{m \cdot n} \). Apply this rule to the innermost expression: \( \left(x^{2}\right)^{2} = x^{2 \cdot 2} = x^{4} \).

Step 2: Apply the Power of a Power Rule Again

Now simplify \( \left(x^{4}\right)^{2} \) using the power of a power rule again: \( \left(x^{4}\right)^{2} = x^{4 \cdot 2} = x^{8} \).

Step 3: Combine the Results

After applying the power of a power rule twice, the expression simplifies completely to \( x^{8} \). This is our final simplified expression.

Key Concepts

Simplifying Exponents

Simplifying exponents can seem tricky at first, but it's all about recognizing and applying rules. In mathematics, an exponent tells us how many times a number, called the base, is multiplied by itself. For example, in the expression \( x^2 \), \( x \) is the base and 2 is the exponent, indicating that \( x \) is multiplied by itself once: \( x \times x \).
To simplify expressions like \( \left( x^2 \right)^2 \), we can use an important tool called the power of a power rule. This rule helps when an exponent is raised to another exponent. By simplifying such expressions, we make equations easier to work with and interpret.

• Breaking down complicated exponent combinations into simpler forms
• Makes solving for variables within equations more manageable
• Allows easier identification of trends or relationships within mathematical models

Whenever you're given nested exponents, remember that simplifying them will ease your calculations and help you reach your solutions faster.

Exponential Notation

Exponential notation is a shorthand way to express repeated multiplication of the same factor. It makes equations and terms more concise and readable. In our example, \( \left[ \left(x^{2}\right)^{2} \right]^{2} \), exponential notation is used to compact multiple layers of multiplication.
Exponential notation is especially useful in fields requiring work with large numbers or complex equations, such as physics or computer science.

• Represents powers of ten very efficiently
• Makes writing complex calculations quick and clear
• Helps maintain accuracy by avoiding errors in counting factors

When using exponential notation, each layer in the chain of exponents can be tackled using known rules, like the power of a power rule, ensuring the transformations are logical and keep the overall relationships unchanged.

Mathematical Simplification

Mathematical simplification is the process of reducing mathematical expressions to their basic form. This makes equations easier to read, understand, and solve. It involves using established rules, like distributing factors, combining like terms, and applying properties of exponents.
In the given exercise, we simplified \( \left[ \left(x^{2}\right)^{2} \right]^{2} \) to get \( x^{8} \). This process involved applying the power of a power rule repeatedly and recognizing patterns. It helps us to shift from convoluted expressions to clear, simple ones.

• Saves time in calculations and reduces potential for errors
• Provides a clear understanding of the problem at hand
• Enables easier manipulation of expressions when solving equations

Simplification is crucial for communicating mathematical ideas effectively—it ensures that we focus on essential numbers and operations, without getting lost in complex, unwieldy expressions.