Problem 72
Question
Simplify by combining like radicals. All variables represent positive real numbers. $$ \sqrt[4]{48}-\sqrt[4]{243}-\sqrt[4]{768} $$
Step-by-Step Solution
Verified Answer
The simplified form is \(-5\sqrt[4]{3}\).
1Step 1: Break Down the Radicals into Prime Factors
Start by breaking down each number inside the radicals into their prime factors.- For \(48\): \(48 = 2^4 \times 3\)- For \(243\): \(243 = 3^5\)- For \(768\): \(768 = 2^8 \times 3\)
2Step 2: Simplify the Root Expressions
Use the properties of fourth roots to simplify:- \(\sqrt[4]{48} = \sqrt[4]{2^4 \times 3} = 2\sqrt[4]{3}\)- \(\sqrt[4]{243} = \sqrt[4]{3^5} = 3\sqrt[4]{3}\)- \(\sqrt[4]{768} = \sqrt[4]{2^8 \times 3} = 2^2\sqrt[4]{3} = 4\sqrt[4]{3}\)
3Step 3: Combine Like Radicals
Add or subtract the coefficients of like radicals (\(\sqrt[4]{3}\) is the like term here):- Expression: \(2\sqrt[4]{3} - 3\sqrt[4]{3} - 4\sqrt[4]{3}\)- Combine: \((2 - 3 - 4)\sqrt[4]{3} = -5\sqrt[4]{3}\)
Key Concepts
Fourth RootsPrime FactorizationLike Radicals
Fourth Roots
When we talk about fourth roots, we're looking at finding a number that, when multiplied by itself four times, gives us the original number. For example, the fourth root of 16 is 2 because \( 2 \times 2 \times 2 \times 2 = 16 \).
When simplifying fourth roots, it's helpful to break down the number inside the radical into its prime factors. This can reveal perfect powers of four.
For instance, with the number 48, its prime factorization is \( 48 = 2^4 \times 3 \). The presence of a whole power of four, such as \( 2^4 \), indicates that we can simplify to \( 2 \) outside the fourth root: \( \sqrt[4]{48} = 2 \sqrt[4]{3} \). This process makes it easier to handle complicated radicals.
When simplifying fourth roots, it's helpful to break down the number inside the radical into its prime factors. This can reveal perfect powers of four.
For instance, with the number 48, its prime factorization is \( 48 = 2^4 \times 3 \). The presence of a whole power of four, such as \( 2^4 \), indicates that we can simplify to \( 2 \) outside the fourth root: \( \sqrt[4]{48} = 2 \sqrt[4]{3} \). This process makes it easier to handle complicated radicals.
Prime Factorization
Prime factorization involves breaking down a number into its prime numbers. It's a fundamental skill, especially when simplifying radicals like fourth roots. Prime numbers are numbers greater than 1, divisible only by 1 and themselves.
For the number 48, the prime factorization process would involve dividing by the smallest prime number, starting with 2:
Utilizing prime factorization helps uncover hidden powers, assisting in simplifying expressions substantially.
For the number 48, the prime factorization process would involve dividing by the smallest prime number, starting with 2:
- 48 divides by 2 to give 24.
- 24 divides by 2 to give 12.
- 12 divides by 2 to give 6.
- 6 divides by 2 to give 3, and 3 is a prime number.
Utilizing prime factorization helps uncover hidden powers, assisting in simplifying expressions substantially.
Like Radicals
Like radicals are similar to like terms in algebra, where you only combine expressions with the same radical part. For example, in the expression \( \sqrt[4]{3} \), you can only combine terms that have \( \sqrt[4]{3} \) as the radical.
When combining like radicals, you handle them similarly to combining like terms.
For example, in the original problem, we simplify: \( 2\sqrt[4]{3} - 3\sqrt[4]{3} - 4\sqrt[4]{3} \). All terms have the common radical part \( \sqrt[4]{3} \). This allows us to combine them by adding the coefficients: \( 2 - 3 - 4 = -5 \).
Thus, the expression simplifies to \( -5\sqrt[4]{3} \). Combining like radicals makes expressions easier to read and understand.
When combining like radicals, you handle them similarly to combining like terms.
For example, in the original problem, we simplify: \( 2\sqrt[4]{3} - 3\sqrt[4]{3} - 4\sqrt[4]{3} \). All terms have the common radical part \( \sqrt[4]{3} \). This allows us to combine them by adding the coefficients: \( 2 - 3 - 4 = -5 \).
Thus, the expression simplifies to \( -5\sqrt[4]{3} \). Combining like radicals makes expressions easier to read and understand.
Other exercises in this chapter
Problem 72
Simplify each expression. All variables represent positive real numbers. See Example 7. $$ \left(\frac{25}{49}\right)^{-3 / 2} $$
View solution Problem 72
Simplify each cube root. See Example \(6 .\) $$ \sqrt[3]{8} $$
View solution Problem 72
Rationalize each denominator. All variables represent positive real numbers. $$ \frac{\sqrt[3]{15 m^{4}}}{\sqrt[3]{12 m^{3}}} $$
View solution Problem 72
Divide. Write all answers in the form \(a+b i\) See Example 8. $$ \frac{2 i}{3+8 i} $$
View solution