Problem 26
Question
Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ \sqrt{4 x^{7} y^{5}}+9 x^{2} \sqrt{x^{3} y^{5}}-5 x y \sqrt{x^{5} y^{3}} $$
Step-by-Step Solution
Verified Answer
The simplified result is \(6x^3y^2\sqrt{xy}\).
1Step 1: Simplify each square root term
Simplify the first term: \( \sqrt{4x^7y^5} \).- Break down the expression under the square root: \( \sqrt{4} = 2 \), and rewrite: \( \sqrt{x^7} = x^3 \sqrt{x} \), \( \sqrt{y^5} = y^2 \sqrt{y} \).- Thus: \( \sqrt{4x^7y^5} = 2x^3y^2\sqrt{xy} \).
2Step 2: Simplify the second term
Simplify the second term: \( 9x^2 \sqrt{x^3y^5} \).- Break down: \( \sqrt{x^3} = x\sqrt{x} \), \( \sqrt{y^5} = y^2\sqrt{y} \).- Thus: \( 9x^2\sqrt{x^3y^5} = 9x^3y^2\sqrt{xy} \).
3Step 3: Simplify the third term
Simplify the third term: \( 5xy \sqrt{x^5y^3} \).- Break down: \( \sqrt{x^5} = x^2\sqrt{x} \), \( \sqrt{y^3} = y\sqrt{y} \).- Thus: \( 5xy\sqrt{x^5y^3} = 5x^3y^2\sqrt{xy} \).
4Step 4: Combine like terms
We now have the expression: \( 2x^3y^2\sqrt{xy} + 9x^3y^2\sqrt{xy} - 5x^3y^2\sqrt{xy} \).- Combine the coefficients: \( (2 + 9 - 5)x^3y^2\sqrt{xy} \).- This simplifies to: \( 6x^3y^2\sqrt{xy} \).
Key Concepts
Adding Radical ExpressionsReal NumbersVariable Exponents
Adding Radical Expressions
Radical expressions often show up as square roots, cube roots, and other roots while solving various mathematical problems. To add radical expressions together, they need to be like radicals. "Like radicals" simply means that the radical expressions have the same index and radicand. Consider like radicals being similar to like terms in algebra, where the numbers or coefficients in front can be combined, but the radical part remains the same.
When working with expressions like \(2\sqrt{3} + 4\sqrt{3}\), since both terms have \(\sqrt{3}\), you can add the coefficients "2" and "4" together to get \(6\sqrt{3}\).
In the problem given, all square roots are simplified to have the same radicand, which is \(\sqrt{xy}\). This means they can be directly added by just handling the coefficients:
When working with expressions like \(2\sqrt{3} + 4\sqrt{3}\), since both terms have \(\sqrt{3}\), you can add the coefficients "2" and "4" together to get \(6\sqrt{3}\).
In the problem given, all square roots are simplified to have the same radicand, which is \(\sqrt{xy}\). This means they can be directly added by just handling the coefficients:
- First term coefficient: 2
- Second term coefficient: 9
- Third term coefficient: -5
Real Numbers
Real numbers are the foundation of most of our computational and theoretical mathematics. This includes both rational numbers, like fractions and integers, and irrational numbers, like \(\pi\) or \(\sqrt{2}\).
When dealing with real numbers in problems like adding radical expressions, it is important to recognize that radicals of non-negative real numbers, like those in the exercise, will always have real solutions. This is because the square root of a negative number isn't considered a real number; instead, it enters the domain of complex numbers.
For instance, in our exercise, we explicitly assume all variables represent positive real numbers. This assumption guarantees that all terms involving square roots will yield real, non-complex solutions—making the solution both more straightforward to simplify and applicable to most practical problems you're likely to encounter at this level of math.
When dealing with real numbers in problems like adding radical expressions, it is important to recognize that radicals of non-negative real numbers, like those in the exercise, will always have real solutions. This is because the square root of a negative number isn't considered a real number; instead, it enters the domain of complex numbers.
For instance, in our exercise, we explicitly assume all variables represent positive real numbers. This assumption guarantees that all terms involving square roots will yield real, non-complex solutions—making the solution both more straightforward to simplify and applicable to most practical problems you're likely to encounter at this level of math.
Variable Exponents
Variable exponents feature prominently in expressions involving powers of variables such as \(x^7\) or \(y^5\). Simplifying radicals with such exponents often requires breaking these expressions down into parts that make the radical simpler to handle.
When faced with something like \(x^7\) under a square root, rewriting it as \(x^6 \cdot x\) allows for easy simplification. The completeness of \(x^6\) enables us to simplify it as \(x^3\) since it is \((x^3)^2\) under the square root, leaving us with \(x^3\sqrt{x}\).
Such simplifications enable us to create like terms in radical expressions. In our practice problem, simplifying each term led to a uniform radicand of \(\sqrt{xy}\), thus achieving a unified expression we could then proceed to add or subtract.
When faced with something like \(x^7\) under a square root, rewriting it as \(x^6 \cdot x\) allows for easy simplification. The completeness of \(x^6\) enables us to simplify it as \(x^3\) since it is \((x^3)^2\) under the square root, leaving us with \(x^3\sqrt{x}\).
Such simplifications enable us to create like terms in radical expressions. In our practice problem, simplifying each term led to a uniform radicand of \(\sqrt{xy}\), thus achieving a unified expression we could then proceed to add or subtract.
- The first term gets simplified to \(2x^3y^2\sqrt{xy}\).
- The second becomes \(9x^3y^2\sqrt{xy}\).
- The third is \(-5x^3y^2\sqrt{xy}\).
Other exercises in this chapter
Problem 26
Find each cube root. $$ \sqrt[3]{27} $$
View solution Problem 26
Use radical notation to rewrite each expression. Simplify if possible. $$ (x-4)^{3 / 4} $$
View solution Problem 26
Use the quotient rule to simplify. Assume that all variables represent positive real numbers. $$ \sqrt{\frac{y^{2} z}{225}} $$
View solution Problem 26
Add or subtract as indicated. Write your answers in the form \(a+b i .\) $$ (-2-4 i)-(6-8 i) $$
View solution