Topic how to simplify a square root: Understanding how to simplify a square root is essential for mastering mathematics. This guide will walk you through the steps and rules needed to break down square roots into their simplest forms, providing clear examples and advanced techniques. Whether you're a student or a math enthusiast, you'll find practical tips and methods to handle any square root simplification with ease. Dive in to learn the foundational concepts and become confident in simplifying square roots!
Table of Content
- Steps to Simplify a Square Root
- Examples
- Simplifying Products of Square Roots
- Simplifying Square Roots of Fractions
- Combining Like Terms
- Understanding Surds
- Examples
- Simplifying Products of Square Roots
- Simplifying Square Roots of Fractions
- Combining Like Terms
- Understanding Surds
- Simplifying Products of Square Roots
- Simplifying Square Roots of Fractions
- Combining Like Terms
- Understanding Surds
- Simplifying Square Roots of Fractions
- Combining Like Terms
- Understanding Surds
- Combining Like Terms
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Steps to Simplify a Square Root
- Factor any perfect squares from the radicand.
- Write the radical expression as a product of radical expressions.
- Simplify the radical expressions.
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Examples
Example 1: Simplify √12
12 can be factored into 4 and 3:
\[\sqrt{12} = \sqrt{4 \times 3}\]
Using the product rule \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\):
\[\sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}\]
Example 2: Simplify √45
45 can be factored into 9 and 5:
\[\sqrt{45} = \sqrt{9 \times 5}\]
Using the product rule:
\[\sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}\]
Example 3: Simplify √8
8 can be factored into 4 and 2:
\[\sqrt{8} = \sqrt{4 \times 2}\]
Using the product rule:
\[\sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}\]
Example 4: Simplify √18
18 can be factored into 9 and 2:
\[\sqrt{18} = \sqrt{9 \times 2}\]
Using the product rule:
\[\sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}\]
Simplifying Products of Square Roots
Example 5: Simplify √6 × √15
Combine the radicals into one expression:
\[\sqrt{6} \times \sqrt{15} = \sqrt{6 \times 15}\]
Factor the product inside the radical:
\[\sqrt{6 \times 15} = \sqrt{2 \times 3 \times 3 \times 5}\]
Group the perfect squares and simplify:
\[\sqrt{2 \times 3^2 \times 5} = 3\sqrt{10}\]
Simplifying Square Roots of Fractions
The quotient rule states that the square root of a quotient is the quotient of the square roots:
\[\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\]
Example 6: Simplify \(\sqrt{\frac{30}{10}}\)
Combine and simplify the fraction under the radical:
\[\sqrt{\frac{30}{10}} = \sqrt{3}\]
Combining Like Terms
Example 7: Simplify \(2\sqrt{12} + 9\sqrt{3}\)
Simplify \(2\sqrt{12}\) first:
\[2\sqrt{12} = 2 \times 2\sqrt{3} = 4\sqrt{3}\]
Now combine with \(9\sqrt{3}\):
\[4\sqrt{3} + 9\sqrt{3} = (4 + 9)\sqrt{3} = 13\sqrt{3}\]
Understanding Surds
A surd is a square root that cannot be simplified further. For example, \(\sqrt{3}\) is a surd, but \(\sqrt{4} = 2\) is not a surd.
Examples
Example 1: Simplify √12
12 can be factored into 4 and 3:
\[\sqrt{12} = \sqrt{4 \times 3}\]
Using the product rule \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\):
\[\sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}\]
Example 2: Simplify √45
45 can be factored into 9 and 5:
\[\sqrt{45} = \sqrt{9 \times 5}\]
Using the product rule:
\[\sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}\]
Example 3: Simplify √8
8 can be factored into 4 and 2:
\[\sqrt{8} = \sqrt{4 \times 2}\]
Using the product rule:
\[\sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}\]
Example 4: Simplify √18
18 can be factored into 9 and 2:
\[\sqrt{18} = \sqrt{9 \times 2}\]
Using the product rule:
\[\sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}\]
Simplifying Products of Square Roots
Example 5: Simplify √6 × √15
Combine the radicals into one expression:
\[\sqrt{6} \times \sqrt{15} = \sqrt{6 \times 15}\]
Factor the product inside the radical:
\[\sqrt{6 \times 15} = \sqrt{2 \times 3 \times 3 \times 5}\]
Group the perfect squares and simplify:
\[\sqrt{2 \times 3^2 \times 5} = 3\sqrt{10}\]
Simplifying Square Roots of Fractions
The quotient rule states that the square root of a quotient is the quotient of the square roots:
\[\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\]
Example 6: Simplify \(\sqrt{\frac{30}{10}}\)
Combine and simplify the fraction under the radical:
\[\sqrt{\frac{30}{10}} = \sqrt{3}\]
Combining Like Terms
Example 7: Simplify \(2\sqrt{12} + 9\sqrt{3}\)
Simplify \(2\sqrt{12}\) first:
\[2\sqrt{12} = 2 \times 2\sqrt{3} = 4\sqrt{3}\]
Now combine with \(9\sqrt{3}\):
\[4\sqrt{3} + 9\sqrt{3} = (4 + 9)\sqrt{3} = 13\sqrt{3}\]
Understanding Surds
A surd is a square root that cannot be simplified further. For example, \(\sqrt{3}\) is a surd, but \(\sqrt{4} = 2\) is not a surd.
Simplifying Products of Square Roots
Example 5: Simplify √6 × √15
Combine the radicals into one expression:
\[\sqrt{6} \times \sqrt{15} = \sqrt{6 \times 15}\]
Factor the product inside the radical:
\[\sqrt{6 \times 15} = \sqrt{2 \times 3 \times 3 \times 5}\]
Group the perfect squares and simplify:
\[\sqrt{2 \times 3^2 \times 5} = 3\sqrt{10}\]
Simplifying Square Roots of Fractions
The quotient rule states that the square root of a quotient is the quotient of the square roots:
\[\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\]
Example 6: Simplify \(\sqrt{\frac{30}{10}}\)
Combine and simplify the fraction under the radical:
\[\sqrt{\frac{30}{10}} = \sqrt{3}\]
Combining Like Terms
Example 7: Simplify \(2\sqrt{12} + 9\sqrt{3}\)
Simplify \(2\sqrt{12}\) first:
\[2\sqrt{12} = 2 \times 2\sqrt{3} = 4\sqrt{3}\]
Now combine with \(9\sqrt{3}\):
\[4\sqrt{3} + 9\sqrt{3} = (4 + 9)\sqrt{3} = 13\sqrt{3}\]
Understanding Surds
A surd is a square root that cannot be simplified further. For example, \(\sqrt{3}\) is a surd, but \(\sqrt{4} = 2\) is not a surd.
Simplifying Square Roots of Fractions
The quotient rule states that the square root of a quotient is the quotient of the square roots:
\[\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\]
Example 6: Simplify \(\sqrt{\frac{30}{10}}\)
Combine and simplify the fraction under the radical:
\[\sqrt{\frac{30}{10}} = \sqrt{3}\]
Combining Like Terms
Example 7: Simplify \(2\sqrt{12} + 9\sqrt{3}\)
Simplify \(2\sqrt{12}\) first:
\[2\sqrt{12} = 2 \times 2\sqrt{3} = 4\sqrt{3}\]
Now combine with \(9\sqrt{3}\):
\[4\sqrt{3} + 9\sqrt{3} = (4 + 9)\sqrt{3} = 13\sqrt{3}\]
Understanding Surds
A surd is a square root that cannot be simplified further. For example, \(\sqrt{3}\) is a surd, but \(\sqrt{4} = 2\) is not a surd.
Combining Like Terms
Example 7: Simplify \(2\sqrt{12} + 9\sqrt{3}\)
Simplify \(2\sqrt{12}\) first:
\[2\sqrt{12} = 2 \times 2\sqrt{3} = 4\sqrt{3}\]
Now combine with \(9\sqrt{3}\):
\[4\sqrt{3} + 9\sqrt{3} = (4 + 9)\sqrt{3} = 13\sqrt{3}\]
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