Problem 63 - 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 in power form is \(3x^{-2}\).

Step 1: Recognize the square root as a power

The square root of any expression can be written as that expression raised to the power of \(\frac{1}{2}\). In this case, \sqrt{\frac{9}{x^{4}}}\ is equivalent to \(\left(\frac{9}{x^{4}}\right)^{\frac{1}{2}}\).

Step 2: Apply the power to both the numerator and the denominator

Use the property of exponents that allows us to apply the power to both the numerator and the denominator separately: \(\left(\frac{9}{x^{4}}\right)^{\frac{1}{2}} = \frac{9^{\frac{1}{2}}}{(x^{4})^{\frac{1}{2}}}\).

Step 3: Simplify the powerful fraction

Compute each part separately: \(9^{\frac{1}{2}}=3\) (because the square root of 9 is 3) and \(x^{4 \times \frac{1}{2}} = x^{2}\) (since multiplying exponents gives \(x^{2}\)). Therefore, \(\frac{9^{\frac{1}{2}}}{(x^{4})^{\frac{1}{2}}} = \frac{3}{x^{2}}\).

Step 4: Express the simplified expression in power form

Write the simplified expression, \(\frac{3}{x^{2}}\), in the power form \(3x^{-2}\). This uses the rule that \(\frac{1}{x^{b}} = x^{-b}\).

Key Concepts

Exponents

Exponents are a fundamental concept in algebra, dealing with repeated multiplication of a number by itself. They are represented as a small number, called the exponent, placed to the upper right of a base number. For example, in the expression \(x^n\), \(x\) is the base and \(n\) is the exponent, indicating that \(x\) is multiplied by itself \(n\) times. When working with exponents, there are several important rules to remember:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \times n}\)
• Power of a Product: \((ab)^n = a^n b^n\)
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\) when \(a eq 0\)

Understanding these rules is crucial, as they allow us to manipulate and simplify expressions involving exponents effectively. When dealing with expressions under a square root, it can also be rewritten using fractional exponents, which ties closely into our next topic.

Simplifying Expressions

Simplifying expressions often involves reducing a complex expression to its simplest form. By doing so, it becomes easier to work with, especially in more extensive calculations or problem solving. The process usually includes applying exponent rules, factoring, and algebraic manipulation.
In the expression \(\sqrt{\frac{9}{x^4}}\), simplifying begins by acknowledging that the square root is equivalent to raising to the power of \(\frac{1}{2}\). This is represented as \(\left(\frac{9}{x^4}\right)^{\frac{1}{2}}\).
Next, this fractional exponent can be distributed across the numerator and the denominator. That results in \(\frac{9^{\frac{1}{2}}}{(x^4)^{\frac{1}{2}}}\). Simplifying each component:

• The square root of 9, or \(9^{\frac{1}{2}}\), simplifies to 3.
• For \(x^4\), the square root means multiplying the exponent by \(\frac{1}{2}\), resulting in \(x^2\).

Thus, the fraction becomes \(\frac{3}{x^2}\). To simplify further into a power form \(ax^b\), use \(x^{-b}\) for terms in the denominator, turning the expression into \(3x^{-2}\). This is a cleaner and more useful form, particularly for integration or derivative calculations.

Fractional Exponents

Fractional exponents are a way of expressing roots, where the exponent is a fraction. The numerator of the fraction represents the power, and the denominator indicates the root. For example, \(a^{\frac{m}{n}}\) implies the \(n\)th root of \(a^m\). Fractional exponents provide a seamless bridge between exponents and roots, which are critical in handling various mathematical problems.
In the exercise \(\sqrt{\frac{9}{x^4}}\), the square root is translated to a fractional exponent, \(\left(\frac{9}{x^4}\right)^{\frac{1}{2}}\). This step is vital in converting more complex radical expressions into exponential form, greatly simplifying the algebraic manipulation involved.
Working with fractional exponents, remember:

• Fractional exponents can apply to both the numerator and denominator separately.
• Ensure that the properties of exponents are used consistently to simplify fully.
• Fractional exponents make it easier to combine and separate terms involving different root operations.

Grasping how fractional exponents function allows for a more comprehensive understanding of algebraic expressions and provides an efficient method for dealing with square, cube roots, and higher orders of roots. This is crucial for solving equations, especially those involving multiple variables or more complex roots.