Problem 42 - Contemporary Precalculus

Question

Simplify the expression without using a calculator. $$\frac{\sqrt{a^{-10} b^{-12}}}{\sqrt{a^{14} d^{-4}}}$$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\frac{\sqrt{a^{-10} b^{-12}}}{\sqrt{a^{14} d^{-4}}}\) without using a calculator. Answer: \(a^2d^2b^6\)

Step 1: Apply the exponent property of square roots

Start by applying the exponent property of square roots, which states that: $$\sqrt{x^y} = x^{\frac{y}{2}}.$$Thus, we get: $$\frac{a^{\frac{-10}{2}} b^{\frac{-12}{2}}}{a^{\frac{14}{2}} d^{\frac{-4}{2}}}$$

Step 2: Simplify the exponents

Now, simplify the exponents by dividing the powers by 2: $$\frac{a^{-5} b^{-6}}{a^7 d^{-2}}$$

Step 3: Apply the properties of exponents

Recall that if we have \(\frac{x^m}{x^n}\), we can rewrite it as \(x^{m-n}\), and if we have \(x^{-n}\), we can rewrite it as \(\frac{1}{x^n}\). Apply these properties to the expression: $$\frac{1}{a^5 b^6} \times \frac{a^7}{d^2}$$

Step 4: Combine terms with the same base

Combine the terms with the same base: $$\frac{a^7}{a^5 d^2 b^6}$$

Step 5: Apply exponent properties again

Apply the exponent properties mentioned in step 3 to get the final simplified expression: $$a^{7-5}d^2b^6 = a^2d^2b^6$$ The simplified expression is: $$\boxed{a^2d^2b^6}$$.

Key Concepts

Square Roots

Understanding square roots is essential in algebra, as they are a type of radical expression that represents the opposite of squaring a number. Essentially, the square root of a number, say \( x \), is a value that, when multiplied by itself, gives \( x \). The notation for the square root of \( x \) is \( \sqrt{x} \).

A fundamental property of square roots is that \( \sqrt{x^y} = x^{\frac{y}{2}} \). This property is derived from the principle that squaring a number and taking its square root are inverse operations. In our exercise, the initial expression had negative exponents inside the square roots. By using this property, we transformed the expression into one involving rational exponents, making it possible to simplify further.

When simplifying radical expressions including square roots, it's important to remember that the goal is to make the expression as straightforward as possible. This often involves rationalizing denominators or combining like terms under the radical. However, in the given exercise, since we deal with variables, we focused on applying the exponent rules to simplify the expression without needing to evaluate the roots directly.

Exponent Properties

Exponent properties, also known as the laws of exponents, are rules that govern the operations on expressions with exponents. When we deal with expressions that have the same base, we can use these properties to simplify the expression. For instance:

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\( x^m \times x^n = x^{m+n} \) states that when multiplying two exponents with the same base, you add the exponents.

\( \frac{x^m}{x^n} = x^{m-n} \) tells us that when dividing exponents with the same base, you subtract the exponents.

\( (x^m)^n = x^{mn} \) explains that when raising a power to another power, you multiply the exponents.

The rule \( x^{-m} = \frac{1}{x^m} \) indicates that a negative exponent represents the reciprocal of the base raised to the opposite positive exponent.

In our exercise, we used the second and the fourth properties. By subtracting the exponents when dividing variables with the same base and converting negative exponents to the reciprocal form, we simplified the complex radical expression significantly.

Properties of Exponents

The properties of exponents are not only limited to multiplication and division but extend to other scenarios as well. Understanding the full range of these properties is vital for simplifying expressions effectively. Some additional exponent properties include:

\Any number (except zero) raised to the zero power is 1, i.e., \( x^0 = 1 \).

The power of a product property \( (xy)^m = x^m \times y^m \) allows us to apply an exponent to a product of bases individually.

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The power of a quotient property \( \left(\frac{x}{y}\right)^m = \frac{x^m}{y^m} \) is similar, but it applies to dividing with a common exponent.

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When simplifying an expression, it's important to use these properties in conjunction, as it might require a combination of them to achieve a fully simplified form.

\In the context of our original exercise, after applying the property that lets us subtract exponents when dividing like bases, we reached a more simplified form that no longer contained negative exponents or radicals. As always with algebraic simplifications, it's essential to pay close attention to the properties in use and apply them correctly to avoid mistakes and ensure the expression is as simple as possible.