Problem 44
Question
For the following problems, simplify each of the radical expressions. $$ -6 \sqrt{72 x^{2} y^{4} z^{10}} $$
Step-by-Step Solution
Verified Answer
Question: Simplify the given radical expression: -6√(72x^2y^4z^10)
Answer: -36√2xyz^5
1Step 1: Factor the expression inside the square root
We will first factor out the greatest common factor inside the square root:
$$
-6 \sqrt{72x^{2}y^{4}z^{10}} = -6\sqrt{2^3\cdot3^2\cdot x^2\cdot y^4\cdot z^{10}}
$$
2Step 2: Simplify the square root using properties of exponents
Now, we will apply the properties of exponents to simplify the square root. Recall that \(\sqrt{a^2} = a\), and \(\sqrt{a^2b^2} = ab\):
$$
-6\sqrt{2^3\cdot3^2\cdot x^2\cdot y^4\cdot z^{10}} = -6 \cdot 3 \cdot \sqrt{2^3}\cdot x\cdot y^2\cdot z^5
$$
3Step 3: Simplify the square root of the remaining term
We are left with the square root of \(2^3\). We will simplify this term by finding the largest perfect square factor and breaking it down:
$$
-6 \cdot 3 \cdot \sqrt{2^3}\cdot x\cdot y^2\cdot z^5 = -6 \cdot 3 \cdot \sqrt{2^2\cdot2}\cdot x\cdot y^2\cdot z^5
$$
Since \(\sqrt{2^2} = 2\), we can continue simplifying the expression:
$$
-6 \cdot 3 \cdot 2\cdot\sqrt{2}\cdot x\cdot y^2\cdot z^5
$$
4Step 4: Multiply coefficients outside the square root
Finally, we will multiply the coefficients outside of the square root:
$$
-6 \cdot 3 \cdot 2\cdot\sqrt{2}\cdot x\cdot y^2\cdot z^5 = -36\sqrt{2}xyz^5
$$
5Step 5: Final Answer
The simplified expression is:
$$
-36\sqrt{2}xyz^5
$$
Key Concepts
Simplifying RadicalsExponentsAlgebra
Simplifying Radicals
Simplifying radicals is all about making radical expressions easier to understand and work with. At its core, simplifying radicals means breaking down the expression inside the square root until you can't simplify any further.
For the problem at hand, we began by looking inside the square root: \( \sqrt{72x^2y^4z^{10}} \). The first step was to factor the number under the radical, in this case, 72. It was broken down into its prime factors: \(72 = 2^3 \cdot 3^2\). This was done to help identify any "perfect squares," which are numbers that, when squared, will give you a whole number (like 4, 9, or 16).
Alongside factoring the numbers, we also need to address the variables \(x^2\), \(y^4\), and \(z^{10}\). Each of these variables can be separated into their exponents that relate to perfect squares. That's because for any variable term \(a^{2n}\), \(\sqrt{a^{2n}} = a^n\). This property allows us to "pull out" perfect squares from under the root, simplifying the expression. Using properties of exponents is crucial to making radical simplification work smoothly.
For the problem at hand, we began by looking inside the square root: \( \sqrt{72x^2y^4z^{10}} \). The first step was to factor the number under the radical, in this case, 72. It was broken down into its prime factors: \(72 = 2^3 \cdot 3^2\). This was done to help identify any "perfect squares," which are numbers that, when squared, will give you a whole number (like 4, 9, or 16).
Alongside factoring the numbers, we also need to address the variables \(x^2\), \(y^4\), and \(z^{10}\). Each of these variables can be separated into their exponents that relate to perfect squares. That's because for any variable term \(a^{2n}\), \(\sqrt{a^{2n}} = a^n\). This property allows us to "pull out" perfect squares from under the root, simplifying the expression. Using properties of exponents is crucial to making radical simplification work smoothly.
Exponents
Exponents are everywhere in algebra and play a pivotal role in simplifying radicals. Exponents tell us how many times a base is multiplied by itself. For instance, \(x^2\) means \(x\) is multiplied by itself to produce \(xx\).
In simplifying radicals, the properties of exponents are handy tools. They allow you to manipulate expressions and make radicals easier to deal with. For example, when simplifying the radicals like in our problem, using the rule \(\sqrt{a^2} = a\) lets us simplify the expression without a hitch.
Exponents also allow for other convenient properties, like \(a^m \cdot a^n = a^{m+n}\) and \((a^m)^n = a^{mn}\). These properties can be particularly useful when trying to consolidate terms and break them down in radical expressions. They work together with factoring to help identify perfect squares and bring terms out from under the radical sign as we saw with \(y^4\) being simplified to \(y^2\), and \(z^{10}\) simplified to \(z^5\).
In simplifying radicals, the properties of exponents are handy tools. They allow you to manipulate expressions and make radicals easier to deal with. For example, when simplifying the radicals like in our problem, using the rule \(\sqrt{a^2} = a\) lets us simplify the expression without a hitch.
Exponents also allow for other convenient properties, like \(a^m \cdot a^n = a^{m+n}\) and \((a^m)^n = a^{mn}\). These properties can be particularly useful when trying to consolidate terms and break them down in radical expressions. They work together with factoring to help identify perfect squares and bring terms out from under the radical sign as we saw with \(y^4\) being simplified to \(y^2\), and \(z^{10}\) simplified to \(z^5\).
Algebra
Algebra acts like a universal language that helps us express mathematical ideas and relationships, such as those seen in simplifying radicals.
When working with radical expressions, algebraic manipulation gives you the tools to break down and combine terms effectively. Solving the given problem involved understanding how each piece of the expression interacts. Recognizing that numbers and variables inside a radical can be reorganized using algebraic laws made it possible to simplify the expression effectively.
Steps in algebra often include factoring, expanding, distributing, and simplifying terms to make the algebraic expressions more understandable. The original expression \(-6 \sqrt{72x^2y^4z^{10}}\) incorporated all these techniques. Factoring helped identify elements to "pull out" from the radical, while multiplying constants brought everything together outside of it.
Although algebra may seem complex, breaking it down into smaller, more digestible tasks can make even the most tricky problems seem manageable. Using algebra effectively can simplify complex expressions and solve equations, making it an essential part of working with radicals and beyond.
When working with radical expressions, algebraic manipulation gives you the tools to break down and combine terms effectively. Solving the given problem involved understanding how each piece of the expression interacts. Recognizing that numbers and variables inside a radical can be reorganized using algebraic laws made it possible to simplify the expression effectively.
Steps in algebra often include factoring, expanding, distributing, and simplifying terms to make the algebraic expressions more understandable. The original expression \(-6 \sqrt{72x^2y^4z^{10}}\) incorporated all these techniques. Factoring helped identify elements to "pull out" from the radical, while multiplying constants brought everything together outside of it.
Although algebra may seem complex, breaking it down into smaller, more digestible tasks can make even the most tricky problems seem manageable. Using algebra effectively can simplify complex expressions and solve equations, making it an essential part of working with radicals and beyond.
Other exercises in this chapter
Problem 44
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Simplify each expression by performing the indicated operation. $$ (3-\sqrt{2})(4-\sqrt{2}) $$
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For the following problems, simplify the expressions. $$ \sqrt{\frac{3}{16}} $$
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