Problem 64 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-80: Simplify the expression, and rationalize the denominator when appropriate. $$ \sqrt{16 a^{8} b^{-2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{4a^4}{b}\).

Step 1: Break Down the Expression Inside the Square Root

The expression inside the square root is \(16a^8b^{-2}\). Begin by separating the integer and variable parts: \(16\), \(a^8\), and \(b^{-2}\).

Step 2: Simplify the Integer Part

The integer \(16\) can be expressed as \(4^2\), which simplifies to \(4\) when taken out of the square root: \(\sqrt{16} = 4\).

Step 3: Simplify the Variable with Exponent Part

Apply the square root to the term \(a^8\): \(\sqrt{a^8} = a^{\frac{8}{2}} = a^4\).

Step 4: Simplify the Negative Exponent

The expression \(b^{-2}\) can be rewritten as \(\frac{1}{b^2}\). Thus, \(\sqrt{b^{-2}} = \sqrt{\frac{1}{b^2}} = \frac{1}{b}\).

Step 5: Combine the Results

Combine the simplified results from the integer and variable parts: \(4a^4 \times \frac{1}{b} = \frac{4a^4}{b}\).

Step 6: Verify Rationalization of the Denominator

In the simplified expression \(\frac{4a^4}{b}\), the denominator is already rational, so no further steps are necessary to rationalize it.

Key Concepts

Rationalize Denominator

Rationalizing a denominator involves eliminating any irrational numbers, such as square roots, from the denominator of a fraction. When you encounter a fraction like \( \frac{1}{\sqrt{b}} \), you want to rationalize it so it becomes a rational number. The idea is to multiply the numerator and the denominator by a conjugate or a form of 1 that will clear the square root from the denominator. For example, multiplying \( \frac{1}{\sqrt{b}} \) by \( \frac{\sqrt{b}}{\sqrt{b}} \) gives you \( \frac{\sqrt{b}}{b} \), which is rationalized.

Rationalizing helps tidy up an expression, making it easier to read and simpler for further computation. If the step-by-step solution already results in a rational denominator, as in the case of \( \frac{4a^4}{b} \), no extra steps are required.

Square Root Simplification

Simplifying square roots means expressing the square root in its simplest form. If a number or expression under the square root can be expressed as the square of another number or expression, it can be simplified. For instance, \( \sqrt{16} \) can be simplified to 4 because 16 equals \( 4^2 \). Similarly, if a variable is raised to an even power under a square root, you can simplify it by halving the exponent.

Consider the expression \( \sqrt{a^8} \). Here, because \( a^8 \) is a perfect square \((a^4 \times a^4)\), you can simplify it to \( a^4 \). Simplifying square root terms makes computations more manageable and expressions less convoluted.

Negative Exponents

Negative exponents indicate that the base should be taken as the reciprocal raised to the positive exponent. For example, \( b^{-2} \) means \( \frac{1}{b^2} \). When you encounter a negative exponent, converting it to a positive exponent fraction is a crucial step to simplifying the expression.

In terms of the provided solution, we handled \( b^{-2} \) by rewriting it as a fraction: \( \frac{1}{b^2} \). This helps set the base to a positive exponent, making it easier to manage operations like multiplication and division. Handling negative exponents this way leads to clearer and more easily solvable expressions.

Variables with Exponents

Variables with exponents represent repeated multiplication of the base variable. For example, \( a^8 \) denotes that the variable \( a \) is multiplied by itself eight times. When simplifying expressions, one can leverage properties of exponents, such as how exponents multiply when applied to powers inside a radical.

In the square root of \( a^8 \), we exploited the property that \( \sqrt{x^m} = x^{\frac{m}{2}} \). This allowed us to easily simplify \( \sqrt{a^8} \) as \( a^4 \). Understanding and applying these properties helps reduce complexity and enhances computational efficiency, both crucial for solving algebraic expressions effectively.