Problem 218
Question
In the following exercises, simplify. $$ 5 \sqrt{12 c^{4}}-3 \sqrt{27 c^{6}} $$
Step-by-Step Solution
Verified Answer
\(\sqrt{3}(10c^2 - 9c^3)\)
1Step 1 - Simplify the first term
Simplify the term inside the square root for the first part: 5\sqrt{12c^4}. This can be broken down as 5\sqrt{4 \cdot 3 \cdot c^4}. Since \(\sqrt{4} = 2\) and \(\sqrt{c^4} = c^2\), this term simplifies to 5 \cdot 2 \cdot c^2 \cdot \sqrt{3} which simplifies further to 10c^2\sqrt{3}.
2Step 2 - Simplify the second term
Simplify the term inside the square root for the second part: -3\sqrt{27c^6}. This can be broken down as -3\sqrt{9 \cdot 3 \cdot c^6}. Since \(\sqrt{9} = 3\) and \(\sqrt{c^6} = c^3\), this term simplifies to -3 \cdot 3 \cdot c^3 \cdot \sqrt{3} which simplifies further to -9c^3\sqrt{3}.
3Step 3 - Combine the simplified terms
Combine the simplified terms: 10c^2\sqrt{3} - 9c^3\sqrt{3}. Since both terms contain \(\sqrt{3}\), factor this out: \(\sqrt{3}(10c^2 - 9c^3)\).
Key Concepts
Square RootsFactoringCombining Like Terms
Square Roots
When simplifying algebraic expressions, one common task is to simplify square roots. The square root of a number is a value that, when multiplied by itself, gives the original number. In the expression \(5 \sqrt{12 c^{4}}-3 \sqrt{27 c^{6}}\), we need to simplify the square roots first.
First, break down the numbers inside the square roots into their prime factors. For example:
\( \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2 \cdot \sqrt{3}\)
This process helps in extracting the square roots of perfect squares (like 4, 9, 16, etc.). For the terms with variables, use the rule \( \sqrt{c^n} = c^{n/2}\). Combining these, we get the simplified terms: \(10c^2\sqrt{3}\) and \(-9c^3\sqrt{3}\).
First, break down the numbers inside the square roots into their prime factors. For example:
\( \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2 \cdot \sqrt{3}\)
This process helps in extracting the square roots of perfect squares (like 4, 9, 16, etc.). For the terms with variables, use the rule \( \sqrt{c^n} = c^{n/2}\). Combining these, we get the simplified terms: \(10c^2\sqrt{3}\) and \(-9c^3\sqrt{3}\).
Factoring
Factoring is another crucial concept in algebra. It involves breaking down expressions into simpler terms that can be multiplied together to get back the original expression. For the given problem, we factor the numbers inside the square roots into their prime components:
For \(12: 12 = 4 \cdot 3\),
For \(27: 27 = 9 \cdot 3\).
This step makes it easier to identify and extract perfect square factors, simplifying the expressions further. For example, \(\sqrt{12 c^4} = \sqrt{4 \cdot 3 \cdot c^4} = 2c^2 \cdot \sqrt{3}\). Such techniques drastically simplify algebraic expressions and make them more manageable.
For \(12: 12 = 4 \cdot 3\),
For \(27: 27 = 9 \cdot 3\).
This step makes it easier to identify and extract perfect square factors, simplifying the expressions further. For example, \(\sqrt{12 c^4} = \sqrt{4 \cdot 3 \cdot c^4} = 2c^2 \cdot \sqrt{3}\). Such techniques drastically simplify algebraic expressions and make them more manageable.
Combining Like Terms
Combining like terms is essential in simplifying algebraic expressions. Like terms are terms whose variables (and their exponents) are the same. In the expression \(10c^2\sqrt{3} - 9c^3\sqrt{3}\), both terms feature \(\sqrt{3}\).
To combine these terms, factor out the common \(\sqrt{3}\) term:
\( 10c^2\sqrt{3} - 9c^3\)\sqrt{3}\
becomes \(\sqrt{3}(10c^2 - 9c^3)\).
This leaves a simpler expression inside the parentheses. Always look for common factors to combine like terms effectively. Simplifying this way helps in better understanding and solving algebraic problems.
To combine these terms, factor out the common \(\sqrt{3}\) term:
\( 10c^2\sqrt{3} - 9c^3\)\sqrt{3}\
becomes \(\sqrt{3}(10c^2 - 9c^3)\).
This leaves a simpler expression inside the parentheses. Always look for common factors to combine like terms effectively. Simplifying this way helps in better understanding and solving algebraic problems.
Other exercises in this chapter
Problem 216
In the following exercises, simplify. $$ 5 \sqrt{x}-8 \sqrt{y} $$
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In the following exercises, simplify. $$ \frac{3}{5} \sqrt{75}-\frac{1}{4} \sqrt{48} $$
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