Problem 9 - Contemporary Precalculus

Question

Simplify the expression. Assume \(a, b, c, d>0\) $$\left(a^{x^{2}}\right)^{1 / x}$$

Step-by-Step Solution

Answer

Answer: The simplified expression for \(\left(a^{x^{2}}\right)^{1 / x}\) is \(a^{x^2-x}\).

Step 1: Apply the power of a power rule

Use the power of a power rule, \((a^m)^n = a^{mn}\). Here, \(m = x^2\) and \(n = \frac{1}{x}\), so we can rewrite the expression as: $$a^{x^2 \cdot \frac{1}{x}}$$

Step 2: Simplify the exponent

Now, we need to simplify the exponent. To do this, let's multiply \(x^2\) by \(\frac{1}{x}\): $$x^2 \cdot \frac{1}{x} = \frac{x^2}{x}$$

Step 3: Simplify further

To simplify the expression even further, we can apply the rule of exponents for division, which states that \(\frac{a^m}{a^n} = a^{m-n}\). In our case, \(m = 2\) and \(n = 1\), so the simplified expression is: $$a^{x^2-x}$$

Step 4: Write the final simplified expression

Now we have simplified the expression to its simplest form. The final answer is: $$\left(a^{x^{2}}\right)^{1 / x} = a^{x^2-x}$$

Key Concepts

Exponentiation Rules

Exponentiation rules are fundamental principles that guide us on how to handle mathematical expressions involving powers. They help us simplify and manipulate expressions efficiently.
These rules include:

• Product of powers: When multiplying two expressions with the same base, you add their exponents: \(a^m \cdot a^n = a^{m+n}\).
• Quotient of powers: When dividing expressions with the same base, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a power: An expression raised to another power involves multiplying the exponents: \((a^m)^n = a^{m \cdot n}\).

Learning these rules allows us to simplify even complex polynomial forms. Simplifying these expressions often makes solving algebraic problems far more manageable.

Power of a Power Rule

The power of a power rule is one of the most useful exponentiation rules for simplifying complex expressions. When you have a power raised to another power, this rule tells us to multiply the exponents.
This is written mathematically as \((a^m)^n = a^{m \cdot n}\). For example, \( (x^3)^2 \) simplifies to \( x^{3 \cdot 2} = x^6 \).
Applying this rule helps us reduce expressions to a more manageable form.

• Look at the base of the expression; if it is the same, multiply the exponents.
• This operation condenses the expression, making calculations easier and revealing a simpler form.

The rule simplifies the seemingly complicated polynomial expressions, transforming them into more solvable forms.

Polynomial Division

Polynomial division involves dividing one polynomial by another, often requiring simplification using exponent rules. It can be daunting at first, but understanding the process makes it much more straightforward.
Division of powers is a key element of this. When dividing powers with the same base, subtract the bottom exponent from the top exponent, like \(\frac{a^m}{a^n} = a^{m-n}\).
For example, \(\frac{x^5}{x^2} = x^{5-2} = x^3\). This operation simplifies the division process and reduces the polynomial's degree.

• Identify like bases in the numerator and denominator.
• Apply the quotient rule by subtracting the exponents.
• Simplify the expression to find the reduced form.

Familiarity with polynomial division and exponent rules provides the tools necessary to tackle complex algebraic operations efficiently.