Question

Simplify each expression. Assume that all variables are positive when they appear. $$\sqrt[4]{162 x^{9} y^{12}}$$

Short answer

\root4{162 x^9 y^{12}} = 3 x^{\frac{9}{4}} y^3 \root4 {2}

Step-by-step solution

- Break Down the Radicand

First, break down the radicand (the expression under the radical) into its prime factors and simplify the variables. Here, we have:\( 162x^9y^{12} \).

- Simplify the Numerical Part

Prime factorize 162. The factorization is:\( 162 = 2 \times 3^4 \). So, the expression with numerical part simplified is:\( \root4 (2 \times 3^4)\times x^9 \times y^{12} \).

- Consider the Variable Part

Simplify the variable parts for the fourth root. Write the expression as:\( \root4 (2 \times 3^4 \times x^9 \times y^{12}) \).

- Apply the Fourth Root

Take the fourth root of each part separately. We may consider: the fourth root of 2 is \( \root4{2}\), then \( 3^4 \times \root4{} = 3\), \( (x^9)^\root4{} = x^{\frac{9}{4}}\), and \( (y^{12})^\root4{} = y^3 \). Combine these results to get:\( 3 x^{\frac{9}{4}} y^3 \)

- Combine Results

Finally, combine the simplified parts to get:\( \root4{162 x^9 y^{12}} = 3 x^{\frac{9}{4}} y^3 \root4 {2} \).

Key concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks, which are prime numbers. A prime number is a number that has no other factors besides 1 and itself. For example:

• The prime factorization of 12 is \(12 = 2^2 \times 3\).
• In our exercise, the prime factorization of 162 is \(162 = 2 \times 3^4\).

Breaking down numbers into primes makes it easier to work with them, especially when they are inside expressions such as radicals.

Variable Exponents

Exponents represent repeated multiplication of the same number or variable. When working with expressions that have variables with exponents, remember these rules:

• Multiplying variables with the same base: \(x^a \times x^b = x^{a+b}\).
• Dividing variables with the same base: \(x^a \div x^b = x^{a-b}\).
• Raising a power to a power: \((x^a)^b = x^{a\cdot b}\).

For our exercise, we're working with exponents inside a radical. For example, \(x^9\) and \(y^{12}\). When applying the fourth root, these become \(x^{\frac{9}{4}}\) and \(y^3\).

Fourth Root

The fourth root of a number is a value that, when multiplied by itself four times, gives the original number. For example, the fourth root of 16 is 2, because \(2 \times 2 \times 2 \times 2 = 16\). In notation, it is represented as \(\root4{}\).
In the given exercise, we find the fourth root of \(162 x^9 y^{12}\). We can simplify the problem by breaking down the number and variable parts separately:

• For the numerical part, \(162 = 2 \times 3^4\).
• For the variable part, \(x^9 = (x^9)^{\frac{1}{4}} = x^{\frac{9}{4}}\) and \(y^{12} = (y^{12})^{\frac{1}{4}} = y^3\).

Finally, the fourth root of \(2\) remains as \(\root4{2}\).

Expression Simplification

Simplifying an expression involves reducing it to its simplest form. In our problem, we started with \(\root4{162 x^9 y^{12}}\). Here are the steps to simplify:

• Factor the numerical part: \(162 = 2 \times 3^4\).
• Break down the variable exponents: \(x^9 = x^{\frac{9}{4}}\) and \(y^{12} = y^3\).
• Apply the fourth root separately to each part: the fourth root of \((3^4)\) is 3, and the fourth root of \(2\) remains \(\root4{2}\).
• Combine the simplified parts: \(3 x^{\frac{9}{4}} y^3 \root4{2}\).

This final expression, \(3 x^{\frac{9}{4}} y^3 \root4{2}\), is the simplest form of the original radical expression.