Problem 39 - Contemporary Precalculus

Question

Simplify the expression without using a calculator. $$\sqrt{16 a^{8} b^{-2}}$$

Step-by-Step Solution

Answer

Answer: The simplified form is $$4a^{4}b^{-1}$$.

Step 1: Find the square root

To find the square root of the given expression, we will take the square root of the number and each term separately. So, the square root of 16 is 4, the square root of $$a^{8}$$ is $$a^{4}$$, and the square root of $$b^{-2}$$ is $$b^{-1}$$.

Step 2: Apply the properties of exponents

Now, let's combine the simplified terms and apply the properties of exponents. $$4a^{4}b^{-1}$$

Step 3: Combine the simplified terms

Since there are no similar terms to combine, the final simplified expression is: $$4a^{4}b^{-1}$$

Key Concepts

Square Root

Understanding the square root is fundamental in simplifying radical expressions. A square root is an operation that asks, 'What number multiplied by itself gives the original number?' For example, the square root of 16, denoted as \( \sqrt{16} \), is 4 because \( 4 \times 4 = 16 \). It's worth noting that every positive number has two square roots, a positive and a negative one (e.g., \( \sqrt{16} \)= 4 or -4), but when simplifying expressions, we typically use the principal (positive) root. Also, when we deal with variables under a square root, we can apply the square root separately to each variable raised to an even exponent, as each variable to an even power is a perfect square.

Furthermore, when simplifying square roots with variables, remember that the square root of a variable to an even power, like \( a^8 \), is the variable raised to half that exponent, which in this case would be \( a^4\) because \( (a^4)^2 = a^8 \). Ensuring you have the basics down, such as these, will make the process of simplifying radical expressions much easier.

Properties of Exponents

The properties of exponents are rules that describe how terms with exponents are handled during multiplication and division. It's important to know these properties when simplifying expressions involving exponents — especially when dealing with radical simplification. The basic properties include:

• Product of powers: \( a^m \times a^n = a^{m+n}\)
• Quotient of powers: \( a^m \div a^n = a^{m-n}\)
• Power of a power: \( (a^m)^n = a^{mn}\)
• Power of a product: \( (ab)^n = a^n b^n\)

By applying these properties, one can manipulate and simplify expressions containing exponents effectively. For example, when we multiply two exponents with the same base, we add the exponents, and when we divide them, we subtract the exponents.

Negative Exponents

Negative exponents can be daunting, but their rule is quite simple. Any term with a negative exponent, such as \( b^{-n} \), signifies a reciprocal process. According to the negative exponent rule, \( b^{-n} = \frac{1}{b^n} \). This means that to simplify an expression with negative exponents, you must take the reciprocal of the term with the negative exponent and change the sign of the exponent to positive.

In practice, when simplifying the square root of \( b^{-2} \) as in our exercise, you would take the reciprocal of \( b^2 \) (which is \( \frac{1}{b^2} \) or \( b^{-2} \)) and then find the square root, which would give you \( b^{-1} \) (or \( \frac{1}{b} \)). This conversion is crucial when you are simplifying radical expressions, as it allows the exponent to come down to a more manageable positive exponent.

Radical Simplification

Radical simplification is the process of breaking down complicated square root or other root expressions into their most reduced form. It often involves a combination of the concepts already discussed: understanding the square root, applying properties of exponents, and handling negative exponents. In our example, simplifying \(\sqrt{16 a^{8} b^{-2}}\) involves several steps. First, we find the square root of the constants and variables separately. Then, we apply exponent properties to combine these terms into a singular expression that has no radicals.

The final step is recognizing when a variable with a negative exponent means that it should be expressed as a reciprocal. Combining these concepts and understanding each step paves the way to effectively simplify radical expressions and achieve a clear, concise result like \(4a^{4}b^{-1}\), where every component is in its simplest form.