Problem 71
Question
Simplify by combining like radicals. All variables represent positive real numbers. $$ \sqrt[4]{32}+5 \sqrt[4]{2}-\sqrt[4]{162} $$
Step-by-Step Solution
Verified Answer
The simplified expression is \( 4\sqrt[4]{2} \).
1Step 1: Break Down Each Radical
First, we simplify each term:- Break down \( \sqrt[4]{32} \): \[ 32 = 2^5 \] Therefore, \( \sqrt[4]{32} = \sqrt[4]{2^4 \times 2} = \sqrt[4]{2^4} \times \sqrt[4]{2} = 2 \sqrt[4]{2} \).- Break down \( \sqrt[4]{162} \): \[ 162 = 81 \times 2 = 3^4 \times 2 \] Therefore, \( \sqrt[4]{162} = \sqrt[4]{3^4 \times 2} = \sqrt[4]{3^4} \times \sqrt[4]{2} = 3 \sqrt[4]{2} \).
2Step 2: Substitute Simplified Radicals
Substitute the simplified radicals back into the expression:\[\sqrt[4]{32} + 5\sqrt[4]{2} - \sqrt[4]{162} = 2\sqrt[4]{2} + 5\sqrt[4]{2} - 3\sqrt[4]{2} \]
3Step 3: Combine Like Radicals
Combine the like radicals by adding or subtracting the coefficients. Add the coefficients of \( \sqrt[4]{2} \): \[2 + 5 - 3 = 4 \]The expression simplifies to:\[4\sqrt[4]{2} \]
4Step 4: Verify the Simplification
Check each component to verify the radicals have been combined correctly. Ensure that all steps were performed accurately and that all radicals with \( \sqrt[4]{2} \) are combined to result in \( 4\sqrt[4]{2} \).
Key Concepts
Simplifying RadicalsLike RadicalsFourth Roots
Simplifying Radicals
Understanding how to simplify radicals is an essential skill in algebra, especially when dealing with expressions that contain roots. A radical is essentially the symbol used to denote roots. In this exercise, we are working with the fourth root, represented as \( \sqrt[4]{x} \), meaning we want the number that, when multiplied by itself four times, gives the value inside the radical.
Simplifying radicals involves breaking down the number inside into a form that allows us to pull out perfect powers. For example, when simplifying \( \sqrt[4]{32} \), consider if any parts of 32 are perfect fourth powers. Recognizing quickly that \( 32 = 2^4 \times 2 \), we pull out 2 since it is within \( \sqrt[4]{2^4} \), simplifying to \( 2 \times \sqrt[4]{2} \).
Using this same logic on \( \sqrt[4]{162} \), after factoring to \( 3^4 \times 2 \), we similarly simplify to \( 3 \times \sqrt[4]{2} \). By understanding these simplification techniques, you're halfway to dealing with any radical expression.
Simplifying radicals involves breaking down the number inside into a form that allows us to pull out perfect powers. For example, when simplifying \( \sqrt[4]{32} \), consider if any parts of 32 are perfect fourth powers. Recognizing quickly that \( 32 = 2^4 \times 2 \), we pull out 2 since it is within \( \sqrt[4]{2^4} \), simplifying to \( 2 \times \sqrt[4]{2} \).
Using this same logic on \( \sqrt[4]{162} \), after factoring to \( 3^4 \times 2 \), we similarly simplify to \( 3 \times \sqrt[4]{2} \). By understanding these simplification techniques, you're halfway to dealing with any radical expression.
Like Radicals
Like radicals are essentially terms that have the same radical part. In the exercise, we deal with fourth roots, specifically the same \( \sqrt[4]{2} \) term appearing in different parts of the expression.
Much like adding and subtracting like terms in algebra (like \( x \) or \( y \)), you can only combine radicals if they have the exact same root and radicand (the number inside the radical).
When an expression such as \( 2\sqrt[4]{2} + 5\sqrt[4]{2} - 3\sqrt[4]{2} \) is presented, you add or subtract only the coefficients, which are the constants in front of the radicals, similar to adding like terms: \( 2 + 5 - 3 \).
Combining like radicals simplifies the expression to \( 4\sqrt[4]{2} \). This method is crucial for efficiently handling radical expressions, making complex-looking problems much simpler.
Much like adding and subtracting like terms in algebra (like \( x \) or \( y \)), you can only combine radicals if they have the exact same root and radicand (the number inside the radical).
When an expression such as \( 2\sqrt[4]{2} + 5\sqrt[4]{2} - 3\sqrt[4]{2} \) is presented, you add or subtract only the coefficients, which are the constants in front of the radicals, similar to adding like terms: \( 2 + 5 - 3 \).
Combining like radicals simplifies the expression to \( 4\sqrt[4]{2} \). This method is crucial for efficiently handling radical expressions, making complex-looking problems much simpler.
Fourth Roots
Fourth roots involve finding a number that, when raised to the fourth power, equals the number inside the radical. The notation \( \sqrt[4]{x} \) is used to denote this operation.
For example, knowing that \( 3^4 = 81 \) shows \( \sqrt[4]{81} = 3 \). Recognizing and breaking numbers into parts that include fourth powers allows simplification, like reducing \( \sqrt[4]{3^4 \times 2} \) to \( 3\sqrt[4]{2} \).
Calculating fourth roots manually involves factorizing the number to see if any sections are perfect fourth powers. If they are, simplify them by removing the root from those parts first, like reducing \( \sqrt[4]{2^4 \times 2} \) to \( 2\sqrt[4]{2} \).
For example, knowing that \( 3^4 = 81 \) shows \( \sqrt[4]{81} = 3 \). Recognizing and breaking numbers into parts that include fourth powers allows simplification, like reducing \( \sqrt[4]{3^4 \times 2} \) to \( 3\sqrt[4]{2} \).
Calculating fourth roots manually involves factorizing the number to see if any sections are perfect fourth powers. If they are, simplify them by removing the root from those parts first, like reducing \( \sqrt[4]{2^4 \times 2} \) to \( 2\sqrt[4]{2} \).
- To simplify calculations, familiarize yourself with small fourth powers like \( 2^4 = 16 \), \( 3^4 = 81 \), etc.
- This allows rapid recognition in exercises and efficient simplifications.
Other exercises in this chapter
Problem 71
Divide. Write all answers in the form a \(+b i.\) $$ \frac{11 i}{4-7 i} $$
View solution Problem 71
Rationalize each denominator. All variables represent positive real numbers. $$ \frac{\sqrt[3]{12 t^{3}}}{\sqrt[3]{54 t^{2}}} $$
View solution Problem 72
Simplify each cube root. See Example 6. $$ \sqrt[3]{8} $$
View solution Problem 72
Solve each equation. Write all proposed solutions. Cross out those that are extraneous. $$ 3 \sqrt{x}=\sqrt{3 x+54} $$
View solution