Problem 43
Question
\(39-48=\) Simplify the expression. $$ \sqrt[3]{108}-\sqrt[3]{32} $$
Step-by-Step Solution
Verified Answer
The simplified expression is \( \sqrt[3]{4} \).
1Step 1: Identify Cube Roots
First, we recognize both terms under the cube roots. We have the expression \( \sqrt[3]{108} - \sqrt[3]{32} \). We need to find the cube roots of 108 and 32.
2Step 2: Simplify Cube Roots
Calculate the cube roots: \( 32 = 2^5 \) which means \( \sqrt[3]{32} = 2^\frac{5}{3} = 2 \cdot \sqrt[3]{4} \). For \( \,\sqrt[3]{108} \), factor 108 to \( 108 = 2^2 \cdot 3^3 \), so \( \sqrt[3]{108} = \sqrt[3]{2^2} \cdot \sqrt[3]{3^3} = 2^{\frac{2}{3}} \cdot 3 = 3 \cdot \sqrt[3]{4} \).
3Step 3: Combine Simplified Roots
After simplification, the expression \( \sqrt[3]{108} - \sqrt[3]{32} \) becomes \( 3\sqrt[3]{4} - 2\sqrt[3]{4} \).
4Step 4: Perform Subtraction
Subtraction of like terms gives us \( (3-2)\sqrt[3]{4} = 1\sqrt[3]{4} \). Thus, the expression simplifies to \( \sqrt[3]{4} \).
Key Concepts
SimplificationFactorizationSubtraction of Like TermsRadicals
Simplification
Simplification is the process of reducing a mathematical expression to its simplest form. We aim to make expressions easier to work with, without changing their value.
In the context of cube roots, simplification involves expressing complicated cube roots in terms of their most basic components. For example, let's take the expression \( \sqrt[3]{108} \). To simplify it, we need to factor 108 into prime factors. This gives us \( 108 = 2^2 \cdot 3^3 \). By breaking down the cube root into simpler parts, we find \( \sqrt[3]{108} = \sqrt[3]{2^2} \cdot \sqrt[3]{3^3} \) which, in turn, becomes \( 2^{\frac{2}{3}} \cdot 3 \).
By analyzing these factors, we can simplify cube root expressions step by step, leading us to easier calculations later on.
In the context of cube roots, simplification involves expressing complicated cube roots in terms of their most basic components. For example, let's take the expression \( \sqrt[3]{108} \). To simplify it, we need to factor 108 into prime factors. This gives us \( 108 = 2^2 \cdot 3^3 \). By breaking down the cube root into simpler parts, we find \( \sqrt[3]{108} = \sqrt[3]{2^2} \cdot \sqrt[3]{3^3} \) which, in turn, becomes \( 2^{\frac{2}{3}} \cdot 3 \).
By analyzing these factors, we can simplify cube root expressions step by step, leading us to easier calculations later on.
Factorization
Factorization is a method of expressing a number as a product of its factors or divisors. It is a fundamental technique used in simplifying expressions involving radicals such as cube roots.
For the expression \( \sqrt[3]{32} \), we first express 32 using its prime factors: \( 32 = 2^5 \). By knowing the factorization, we can then represent the cube root as \( 2^{\frac{5}{3}} \), simplifying further to \( 2 \cdot \sqrt[3]{4} \).
Similarly, for the cube root of 108, we factor it to \( 108 = 2^2 \cdot 3^3 \), which facilitates finding the cube root \( \sqrt[3]{108} = \sqrt[3]{2^2} \cdot \sqrt[3]{3^3} \) simplifying it to \( 2^{\frac{2}{3}} \cdot 3 \) or equivalently \( 3 \cdot \sqrt[3]{4} \).
Factorization helps break down complicated expressions into segments that are much easier to handle and simplify.
For the expression \( \sqrt[3]{32} \), we first express 32 using its prime factors: \( 32 = 2^5 \). By knowing the factorization, we can then represent the cube root as \( 2^{\frac{5}{3}} \), simplifying further to \( 2 \cdot \sqrt[3]{4} \).
Similarly, for the cube root of 108, we factor it to \( 108 = 2^2 \cdot 3^3 \), which facilitates finding the cube root \( \sqrt[3]{108} = \sqrt[3]{2^2} \cdot \sqrt[3]{3^3} \) simplifying it to \( 2^{\frac{2}{3}} \cdot 3 \) or equivalently \( 3 \cdot \sqrt[3]{4} \).
Factorization helps break down complicated expressions into segments that are much easier to handle and simplify.
Subtraction of Like Terms
The operation of subtracting like terms is a crucial step in simplifying expressions, especially when dealing with radicals.
In our original expression, \( \sqrt[3]{108} - \sqrt[3]{32} \), simplifying both cube roots brings us to a stage where the terms share a common radical part, specifically \( \sqrt[3]{4} \).
By expressing \( \sqrt[3]{108} \) as \( 3\sqrt[3]{4} \) and \( \sqrt[3]{32} \) as \( 2\sqrt[3]{4} \), these terms can be directly subtracted because they have like radicals. This process, referred to as the subtraction of like terms, entails simplifying:
In our original expression, \( \sqrt[3]{108} - \sqrt[3]{32} \), simplifying both cube roots brings us to a stage where the terms share a common radical part, specifically \( \sqrt[3]{4} \).
By expressing \( \sqrt[3]{108} \) as \( 3\sqrt[3]{4} \) and \( \sqrt[3]{32} \) as \( 2\sqrt[3]{4} \), these terms can be directly subtracted because they have like radicals. This process, referred to as the subtraction of like terms, entails simplifying:
- Identifying the shared radical, \( \sqrt[3]{4} \)
- Subtracting the coefficients: \( 3 - 2 = 1 \)
- Resulting in a simpler term: \( 1\sqrt[3]{4} \)
Radicals
Radicals are expressions involving roots, such as square roots or cube roots. Radicals present a method for expressing roots of numbers in a general form.
Cube roots, expressed as \( \sqrt[3]{x} \), are particularly interesting because they represent a number that, when multiplied three times, gives \( x \). In the example \( \sqrt[3]{108} - \sqrt[3]{32} \), we deal specifically with cube roots.
To handle radicals effectively:
Cube roots, expressed as \( \sqrt[3]{x} \), are particularly interesting because they represent a number that, when multiplied three times, gives \( x \). In the example \( \sqrt[3]{108} - \sqrt[3]{32} \), we deal specifically with cube roots.
To handle radicals effectively:
- Understand the radical's index (3 for cube roots)
- Simplify the inside using factorization
- Identify common radicals when working with addition or subtraction
Other exercises in this chapter
Problem 43
Factor the expression completely. $$ 2 x^{2}+5 x+3 $$
View solution Problem 43
Express the inequality in interval notation, and then graph the corresponding interval. $$ x>-1 $$
View solution Problem 43
Multiply the algebraic expressions using the FOIL method, and simplify. \((3 x+5)(2 x-1)\)
View solution Problem 43
Mary’s backyard vegetable garden measures 20 \(\mathrm{ft}\) by 30 \(\mathrm{ft}\) , so its area is \(20 \times 30=600 \mathrm{ft}^{2}\). She decides to make it
View solution