Problem 41 - Contemporary Precalculus

Question

Simplify the expression without using a calculator. $$\frac{\sqrt{c^{2} d^{6}}}{\sqrt{4 c^{3} d^{-4}}}$$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is $$\frac{1}{2} c^{-\frac{1}{2}} d^5$$.

Step 1: Apply the properties of square roots

The square root of a product is the product of the square roots, so we can break up the numerator and denominator as follows: $$\frac{\sqrt{c^{2} d^{6}}}{\sqrt{4 c^{3} d^{-4}}} = \frac{\sqrt{c^2} \sqrt{d^6}}{\sqrt{4} \sqrt{c^3} \sqrt{d^{-4}}}$$

Step 2: Apply the properties of exponents

Recall that the square root of a number is the same as raising that number to the power 1/2: $$\frac{\sqrt{c^2} \sqrt{d^6}}{\sqrt{4} \sqrt{c^3} \sqrt{d^{-4}}} = \frac{c^{(2\cdot \frac{1}{2})} d^{(6\cdot \frac{1}{2})}}{ 4^{\frac{1}{2}} c^{\frac{1}{2}\cdot3} d^{\frac{1}{2}\cdot(-4)}}$$

Step 3: Simplify the exponents

Multiply the powers for each term: $$\frac{c^{(2\cdot \frac{1}{2})} d^{(6\cdot \frac{1}{2})}}{ 4^{\frac{1}{2}} c^{\frac{1}{2}\cdot3} d^{\frac{1}{2}\cdot(-4)}} = \frac{c^1 d^3}{ 4^{\frac{1}{2}} c^{\frac{3}{2}} d^{-2}}$$

Step 4: Apply the division property of exponents

When dividing with the same base, subtract the exponents: $$\frac{c^1 d^3}{ 4^{\frac{1}{2}} c^{\frac{3}{2}} d^{-2}} = \frac{1}{4^{\frac{1}{2}}} c^{(1-\frac{3}{2})} d^{(3-(-2))}$$ $$\frac{1}{\sqrt 4} c^{-\frac{1}{2}} d^5$$

Step 5: Simplify the expression

The square root of 4 is 2, so the final simplified expression is: $$\frac{1}{2} c^{-\frac{1}{2}} d^5$$

Key Concepts

Properties of Square Roots

Understanding the properties of square roots is crucial when simplifying expressions that involve radical symbols. The square root of a number represents a value which, when multiplied by itself, yields the original number. For instance, the square root of 4 is 2 because 2 multiplied by 2 equals 4.

One key property is that the square root of a product is the product of the square roots of the factors: \[ \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \.\] Another important point to remember is that the square root of a quotient is the quotient of the square roots of the numerator and the denominator: \[ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \.\] These properties allow us to break down and simplify complex square root expressions. In the given exercise, applying these properties helps us to separate the components within the square roots before further simplification.

Properties of Exponents

The rules that govern how to manipulate properties of exponents are pivotal for simplifying expressions. Exponents, also known as powers, are shorthand for repeated multiplication of the same factor. When you have an expression like \( c^2 \), it means \( c \times c \).

Some integral properties include:

• The product of powers property: \( a^m \cdot a^n = a^{m+n} \) for multiplying like bases,
• The power of a power property: \( (a^m)^n = a^{m \cdot n} \) for raising a power to another power,
• The power of a product property: \( (a \cdot b)^n = a^n \cdot b^n \) which tells us how to raise a product to a power.

Applying these properties allows us to simplify the exponent portions of the expression in the exercise, such as recognizing \( \sqrt{c^2} \) as \( c^{(2 \cdot \frac{1}{2})} \) due to the power of a power property.

Division Property of Exponents

The division property of exponents comes into play when dividing terms with the same base. Simply put, to divide powers with the same base, we subtract the exponents: \[ \frac{a^m}{a^n} = a^{m-n} \.\] This rule is extremely helpful when we need to simplify the division of exponential terms, as demonstrated in the given exercise.

When we observe the expression \( \frac{c^1 d^3}{ 4^{\frac{1}{2}} c^{\frac{3}{2}} d^{-2}} \), we apply this property, resulting in \( c^{(1-\frac{3}{2})} \) and \( d^{(3-(-2))} \), which streamlines the expression to \( c^{-\frac{1}{2}} d^5 \) after simplifying. Note that it's important to be careful with negative exponents as they indicate the reciprocal of the base raised to the positive exponent.