Problem 63 - Brief Applied Calculus

Question

Write each expression in power form \(a x^{b}\) for numbers \(a\) and \(b\). $$ \sqrt{\frac{9}{x^{4}}} $$

Step-by-Step Solution

Answer

The expression is equivalent to \(3x^{-2}\).

Step 1: Understand the Expression

We begin with the expression \(\sqrt{\frac{9}{x^{4}}}\). This expression involves a square root of a fraction.

Step 2: Convert the Square Root to a Power

Recall that the square root of a number can be written as raising that number to the power of \(\frac{1}{2}\). Thus, \(\sqrt{\frac{9}{x^{4}}}\) is equivalent to \(\left(\frac{9}{x^{4}}\right)^{\frac{1}{2}}\).

Step 3: Apply the Power to Both Numerator and Denominator

The property \( \left(\frac{a}{b}\right)^{n} = \frac{a^n}{b^n} \) allows us to split the expression: \( \left(\frac{9}{x^{4}}\right)^{\frac{1}{2}} \) becomes \( \frac{9^{\frac{1}{2}}}{(x^{4})^{\frac{1}{2}}} \).

Step 4: Simplify the Numerator

Calculate \(9^{\frac{1}{2}}\), which is the square root of 9, resulting in 3.

Step 5: Simplify the Denominator

Use the property \((x^m)^{n} = x^{mn}\) to simplify \((x^{4})^{\frac{1}{2}}\) to \(x^{2}\).

Step 6: Form the Power Expression

Combine the simplified parts: the expression \(\frac{3}{x^{2}}\) is equivalent to \(3x^{-2}\) by the property \(\frac{1}{x^b} = x^{-b}\).

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of a number by itself. When you see a number like \( x^4 \), it means \( x \), multiplied by itself 4 times: \( x \times x \times x \times x \).

Exponents have several key properties that make them very useful in algebra:

• Product of Powers: \( x^a \times x^b = x^{a+b} \)
• Power of a Power: \( (x^a)^b = x^{a \cdot b} \)
• Quotient of Powers: \( \frac{x^a}{x^b} = x^{a-b} \)
• Zero Exponent: Any non-zero number to the power of 0 is 1, so \( x^0 = 1 \)

Exponents make expressions more concise and simplify calculations, especially when working with larger numbers or variables. Understanding exponents is essential in transitioning expressions into different forms, such as with power forms observed in this exercise.

Fractional Powers

Fractional powers, also known as rational exponents, are a way to express roots using exponents. The general rule is that \( a^{\frac{m}{n}} \) is equivalent to \( \sqrt[n]{a^m} \), which combines both the process of raising a base to a power and taking the root.

Here are some examples for clarity:

• \( a^{\frac{1}{2}} \) is the same as the square root of \( a \), \( \sqrt{a} \)
• \( a^{\frac{1}{3}} \) represents the cube root of \( a \), \( \sqrt[3]{a} \)
• \( a^{\frac{3}{2}} \) can be interpreted as the square root of \( a^3 \).

When working with fractional powers, applicating the correct power to each part of a fraction or multiplied expression is vital, like in our problem where we treated both the numerator and the denominator with the square root property of \( \frac{1}{2} \).

Simplification

Simplification is the process of making an algebraic expression easier to understand or work with. This often involves reducing expressions to their simplest form.

The simplification process in this exercise, breaking down \( \sqrt{\frac{9}{x^4}} \) illustrates several key techniques:

• Converting square roots to fractional powers: using \( \frac{1}{2} \) as the exponent
• Applying power properties such as \( (x^m)^{n} = x^{m\cdot n} \)
• Breaking apart power expressions across fraction components: \( \left(\frac{a}{b}\right)^{n} = \frac{a^n}{b^n} \)
• Reducing or combining terms where possible, like expressing \( \frac{3}{x^2} \) as \( 3x^{-2} \)

The ultimate goal is to achieve an expression that is both accurate and as straightforward as possible, often using the fewest terms.

Power Expressions

Power expressions involve using exponents to represent algebraic expressions. They allow for compact representation and easier manipulation of expressions. For instance, turning \( \sqrt{\frac{9}{x^4}} \) into a simpler power form such as \( 3x^{-2} \).

When dealing with power expressions, understanding these core ideas is crucial:

• Using Exponent Rules: Use properties such as \( (x^m)^n = x^{mn} \) to simplify expressions effectively.
• Transitioning between Forms: Using equivalent forms to move between expressions involving roots and their corresponding fractional powers.
• Negative Exponents: Interpret \( x^{-b} \) as \( \frac{1}{x^b} \), showing the reciprocal relationship, which is fundamental in our exercise.

Power expressions are central in algebra as they streamline operations by reducing the complexity of expressions, making further algebraic operations more straightforward.