Problem 82
Question
Simplify the expression. $$ \frac{1}{2} x^{-1 / 2}(x+y)^{1 / 3}+\frac{1}{3} x^{1 / 2}(x+y)^{-2 / 3} $$
Step-by-Step Solution
Verified Answer
\(x^{-\frac{1}{2}}(x+y)^{-\frac{2}{3}}\left(\frac{5}{6}x+\frac{1}{2}y+\frac{1}{3}x^{\frac{5}{2}}(x+y)\right)\)
1Step 1: Rewrite the Expression
Given expression:
\[
\frac{1}{2}x^{-\frac{1}{2}}(x+y)^{\frac{1}{3}}+\frac{1}{3}x^{\frac{1}{2}}(x+y)^{-\frac{2}{3}}
\]
2Step 2: Factor out Common Factors
We notice that both terms have common factors of \(x\) and \((x+y)\). Let's factor out the lower exponent of each variable from both terms:
\[
x^{-\frac{1}{2}}(x+y)^{-\frac{2}{3}}\left(\frac{1}{2}(x+y)^{\frac{3}{3}}+\frac{1}{3}x^{\frac{3}{2}}\right)
\]
3Step 3: Simplify the Exponents
Now we simplify the exponents by adding the exponents of the terms inside the bracket:
\[
x^{-\frac{1}{2}}(x+y)^{-\frac{2}{3}}\left(\frac{1}{2}(x+y)+\frac{1}{3}x(x+y)\right)
\]
4Step 4: Distribute the Terms
Next, we distribute the terms inside the bracket:
\[
x^{-\frac{1}{2}}(x+y)^{-\frac{2}{3}}\left(\frac{1}{2}x+\frac{1}{2}y+\frac{1}{3}x^{\frac{5}{2}}(x+y)\right)
\]
5Step 5: Combine Like Terms
Now let's combine the like terms inside the bracket:
\[
x^{-\frac{1}{2}}(x+y)^{-\frac{2}{3}}\left(\frac{5}{6}x+\frac{1}{2}y+\frac{1}{3}x^{\frac{5}{2}}(x+y)\right)
\]
Now the expression is in its simplified form:
\[
x^{-\frac{1}{2}}(x+y)^{-\frac{2}{3}}\left(\frac{5}{6}x+\frac{1}{2}y+\frac{1}{3}x^{\frac{5}{2}}(x+y)\right)
\]
Key Concepts
Exponents and PowersFactoring Algebraic ExpressionsCombining Like Terms
Exponents and Powers
When dealing with exponents and powers, the key is to understand how to manipulate these numbers. In algebra, powers denote repeated multiplication and are written as a base number raised to an exponent or power. For example, in the expression \(x^3\), \(x\) is the base and \(3\) is the exponent, meaning that \(x\) is multiplied by itself 3 times.
Rewriting expressions with negative or fractional exponents can often simplify the solving process. A negative exponent, such as \(x^{-1}\), means to take the reciprocal of the base, which in this case would be \(\frac{1}{x}\). Fractional exponents, like \(x^{1/3}\), represent roots; in this instance, \(x^{1/3}\) is the cube root of \(x\).
In the original exercise, powers with fractional exponents are simplified by recognizing equivalent expressions. Understanding that terms can be rewritten using the properties of exponents is crucial, such as \((x+y)^{3/3} = (x+y)^1 = x+y\). This step is what makes the process of combining like terms possible later on.
Rewriting expressions with negative or fractional exponents can often simplify the solving process. A negative exponent, such as \(x^{-1}\), means to take the reciprocal of the base, which in this case would be \(\frac{1}{x}\). Fractional exponents, like \(x^{1/3}\), represent roots; in this instance, \(x^{1/3}\) is the cube root of \(x\).
In the original exercise, powers with fractional exponents are simplified by recognizing equivalent expressions. Understanding that terms can be rewritten using the properties of exponents is crucial, such as \((x+y)^{3/3} = (x+y)^1 = x+y\). This step is what makes the process of combining like terms possible later on.
Factoring Algebraic Expressions
Factoring plays a significant role in simplifying algebraic expressions. It involves breaking down a complex expression into simpler factors that, when multiplied together, give back the original expression. This is similar to finding what numbers can be multiplied to reach a given product in arithmetic.
To factor an expression, look for common factors in each term. For instance, if each term of an expression includes a variable to a certain power, you can factor out the lowest power of that variable. This step is seen in the original exercise where the lowest exponents of \(x\) and \((x+y)\) are factored out to streamline the expression. The expression \(x^{-1/2}(x+y)^{-2/3}\) is separated as a common factor, simplifying the consequent steps of the simplification process.
The ability to recognize and extract common factors paves the way to restructure expressions and is a precursor to further simplification, such as combining like terms.
To factor an expression, look for common factors in each term. For instance, if each term of an expression includes a variable to a certain power, you can factor out the lowest power of that variable. This step is seen in the original exercise where the lowest exponents of \(x\) and \((x+y)\) are factored out to streamline the expression. The expression \(x^{-1/2}(x+y)^{-2/3}\) is separated as a common factor, simplifying the consequent steps of the simplification process.
The ability to recognize and extract common factors paves the way to restructure expressions and is a precursor to further simplification, such as combining like terms.
Combining Like Terms
After using exponents correctly and factoring out common terms, the next essential skill is combining like terms. Like terms are terms that have the exact same variable raised to the same power. They can be summed or subtracted from each other to condense the expression.
For instance, in the exercise, after distributing and breaking down the terms, we identify like terms that can be combined. The expression inside the bracket \(\left(\frac{1}{2}x + \frac{1}{2}y + \frac{1}{3}x^{5/2}(x+y)\right)\) contains like terms involving \(x\), which are \(\frac{1}{2}x\) and the \(x\) in the term \(\frac{1}{3}x^{5/2}(x+y)\). By combining the coefficients (the numerical parts) of these terms, we simplify the expression further, making it more manageable and easier to understand.
Mastering the skill of combining like terms is important not just for simplification, but also for solving equations and inequalities, where simplification is often the first step.
For instance, in the exercise, after distributing and breaking down the terms, we identify like terms that can be combined. The expression inside the bracket \(\left(\frac{1}{2}x + \frac{1}{2}y + \frac{1}{3}x^{5/2}(x+y)\right)\) contains like terms involving \(x\), which are \(\frac{1}{2}x\) and the \(x\) in the term \(\frac{1}{3}x^{5/2}(x+y)\). By combining the coefficients (the numerical parts) of these terms, we simplify the expression further, making it more manageable and easier to understand.
Mastering the skill of combining like terms is important not just for simplification, but also for solving equations and inequalities, where simplification is often the first step.
Other exercises in this chapter
Problem 81
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Simplify the expression. $$ \frac{\frac{1}{2}\left(1+x^{1 / 3}\right) x^{-1 / 2}-\frac{1}{3} x^{1 / 2} \cdot x^{-2 / 3}}{\left(1+x^{1 / 3}\right)^{2}} $$
View solution