Problem 75 - Intermediate Algebra

Question

Use the quotient rule to divide. Then simplify if possible. See Example 5 $$ \frac{\sqrt[5]{64 x^{10} y^{3}}}{\sqrt[5]{2 x^{3} y^{-7}}} $$

Step-by-Step Solution

Answer

The simplified form is \( 2 x^{7/5} y^2 \).

Step 1: Apply the Quotient Rule for Roots

To divide two radicals with the same index, we apply the quotient rule which states that \( \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}} \). In this case, we have fifth roots: \[ \frac{\sqrt[5]{64 x^{10} y^{3}}}{\sqrt[5]{2 x^{3} y^{-7}}} = \sqrt[5]{\frac{64 x^{10} y^{3}}{2 x^{3} y^{-7}}} \]

Step 2: Simplify the Fraction Inside the Root

Simplify the fraction inside the fifth root by dividing the coefficients and exponents. - Divide the coefficients: \( \frac{64}{2} = 32 \) - Subtract the exponents for \( x \): \( 10 - 3 = 7 \) - Subtract the exponents for \( y \): \( 3 - (-7) = 3 + 7 = 10 \)Thus, the fraction becomes: \[ \frac{64 x^{10} y^{3}}{2 x^{3} y^{-7}} = 32 x^7 y^{10} \]

Step 3: Take the Fifth Root of the Simplified Expression

Now, apply the fifth root to each part of the simplified expression: \[ \sqrt[5]{32 x^7 y^{10}} \]- Calculate the fifth root of 32: \( 32 = 2^5 \) so \( \sqrt[5]{32} = 2 \)- Apply the fifth root to \( x^7 \): \( \sqrt[5]{x^7} = x^{7/5} \)- Apply the fifth root to \( y^{10} \): \( \sqrt[5]{y^{10}} = y^{2} \)

Step 4: Write the Final Expression

Combine the simplified terms after taking the fifth root. The result is: \[ 2 x^{7/5} y^2 \]

Key Concepts

Simplifying Radicals

Simplifying radicals involves reducing the expression under the root to its simplest form. This process often means breaking down numbers and variables to easily manage them under the radical sign. For radicals, you search for perfect powers, in this case, perfect fifth powers because the index is 5. For instance, if the expression involves simplifying \( \sqrt[5]{32} \), you would identify 32 as a perfect fifth power because it is equal to \( 2^5 \).

• Identify and divide coefficients under the radicals.
• Subtract exponents of like variables.
• Break down the expression to simplify further.

In this example, simplifying the fraction inside the radical also involves reducing \( x^{10} \) and \( x^3 \) by subtracting exponents to get \( x^7 \). Similarly, subtracting \( y^{-7} \) from \( y^3 \) gives \( y^{10} \). This simplification prepares the expression to have its radical root calculated efficiently.

Fifth Roots

Fifth roots are specific types of roots with the index of 5, meaning you're looking for a number that when multiplied by itself five times will yield the original number. Finding these often involves knowing the fundamental tactics to break down numbers into base powers. For example, when you encounter the expression \( \sqrt[5]{32} \), you're asked to find the number that results in 32 after being raised to the fifth power, which is 2.

• Recognize exponent patterns to deduce fifth roots.
• Express numbers in terms of fifth powers where possible.

In algebraic terms, fifth roots involving variables like in \( x^{7/5} \) mean taking each variable's power and dividing it by the fifth root, which sometimes results in fractional exponents. This step is crucial when simplifying complex expressions that originate from dividing and simplifying their radical components.

Exponent Rules

Exponent rules are essential for successfully handling problems involving radicals, especially when simplifying expressions. These rules allow you to manipulate expressions with exponents to make calculations more straightforward. By understanding these rules, you simplify expressions like \( x^{10} \) and \( x^{3} \) to \( x^7 \) by subtracting exponents, and transform \( y^{3} \) and \( y^{-7} \) into \( y^{10} \).

• Subtract exponents when dividing similar bases.
• Add exponents when multiplying similar bases.
• Raise powers to new powers by multiplying exponents.

Using exponent rules is crucial when simplifying radical expressions, efficiently managing the arithmetic processes involved in the simplification so you can focus on finding real numbers and solving the roots.

Radical Expressions

Radical expressions include any algebraic expressions that contain roots. The key to working with these is transforming and simplifying them whenever possible to relieve complexity and make calculations easier. In our example, we start with a quotient under two separate radical fifth roots and need to transform it into a single radical expression with the quotient rule.

• Combine divisors under a single radical for simplification.
• Apply relevant rules to simplify expressions under each root.
• Evaluate for any fractions and simplify them.

After simplifying the fraction \( \frac{64 x^{10} y^{3}}{2 x^{3} y^{-7}} \), we reduce it to \( 32 x^7 y^{10} \). Transforming and then simplifying radical expressions is a strategic approach to break down complex problems into manageable steps, always aiming for the simplest form possible.