Properties of Radicals and Rationalizing Denominators
Your calculator says $\sqrt{72} \approx 8.4852813742$, but the answer key says $6\sqrt{2}$. Are they the same number? And why would a key write $\dfrac{\sqrt{2}}{\sqrt{3}+\sqrt{2}}$ as $\sqrt{6}-2$? In this section you learn the moves that put any radical expression into its simplest form, so you can compare answers, add them, and keep working with them.
Simplifying: Pulling Out Factors
A radical is in simplified form when there is no perfect-power factor left under the root. For a square root, that means no perfect square factor besides 1. For a cube root, no perfect cube, and so on.
Useful perfect powers to recognize:
Type
Values
Squares
4, 9, 16, 25, 36, 49, 64, 81, 100
Cubes
8, 27, 64, 125, 216
Fourth powers
16, 81, 256
Variables work the same way (assume the variables are positive). Divide each exponent by the index: the quotient tells you how many copies come out, and the remainder stays in.
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Extract all possible factors from the following radicals: a) $\sqrt{72}$ b) $\sqrt[3]{250}$
Extract all possible factors from the following radicals:
Radicals are powers with fractional exponents: $\sqrt[n]{a^m} = a^{m/n}$. That makes two new moves easy. You can reduce the index by simplifying the fraction $m/n$, and you can combine nested roots by multiplying fractions.
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Simplify the following radicals by dividing the index and the exponent by their greatest common factor: a) $\sqrt[6]{27}$ b) $\sqrt[8]{16}$
Simplify the following radicals by dividing the index and the exponent by their greatest common factor:
Simplify the following expression by introducing the factors into the innermost radical and expressing it as a single root: $\sqrt{2 \cdot \sqrt[3]{2}}$
Simplify the following expression by introducing the factors into the innermost radical and expressing it as a single root:
Like radicals have the same index and the same radicand. You can add or subtract them just as you combine like terms: $3x + 2x = 5x$, so $3\sqrt{2}+2\sqrt{2} = 5\sqrt{2}$.
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Perform the following additions and subtractions of like radicals: a) $5\sqrt{3} - 2\sqrt{3} + 4\sqrt{3}$ b) $\frac{1}{2}\sqrt{7} + \frac{3}{2}\sqrt{7} - \sqrt{7}$
Perform the following additions and subtractions of like radicals:
a) $5\sqrt{3} - 2\sqrt{3} + 4\sqrt{3}$
b) $\frac{1}{2}\sqrt{7} + \frac{3}{2}\sqrt{7} - \sqrt{7}$
With the same index, multiply or divide the radicands directly: $\sqrt[n]{a}\cdot\sqrt[n]{b} = \sqrt[n]{ab}$. With different indices, first rewrite everything with a common index.
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Perform the following multiplications of radicals with the same index and extract factors if possible: a) $\sqrt{2} \cdot \sqrt{8}$ b) $\sqrt[3]{4} \cdot \sqrt[3]{10}$
Perform the following multiplications of radicals with the same index and extract factors if possible:
Leaving a radical in a denominator is not "wrong," but it is not standard form. Rationalizing means multiplying the top and bottom by whatever makes the denominator free of radicals. The value doesn't change, because you are multiplying by $1$.
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Rationalize the denominator of the following fractions and simplify the result: a) $\frac{6}{\sqrt{3}}$ b) $\frac{10}{\sqrt{5}}$
Rationalize the denominator of the following fractions and simplify the result:
If the denominator has two terms, like $\sqrt{a}+\sqrt{b}$, multiplying by $\sqrt{a}$ doesn't clear both radicals. Use the conjugate: the same two terms with the opposite sign in the middle.
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Rationalize the following expressions by multiplying by the conjugate of the denominator: a) $\frac{4}{\sqrt{5}-1}$ b) $\frac{2}{\sqrt{3}+1}$
Rationalize the following expressions by multiplying by the conjugate of the denominator:
Radicals show up whenever you use the Pythagorean Theorem or a formula with a square root. Exact answers keep every digit, and simplified radicals make them readable.
In an isosceles triangle the height lands on the middle of the base. That creates a right triangle with legs $h$ and $\tfrac{b}{2}$, and hypotenuse equal to one of the equal sides.
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The period $T$ of oscillation of a pendulum is calculated with the formula $T = 2\pi \sqrt{\frac{l}{g}}$. If we have a pendulum of length $l = 12$ feet and we consider…
The period $T$ of oscillation of a pendulum is calculated with the formula $T = 2\pi \sqrt{\frac{l}{g}}$. If we have a pendulum of length $l = 12$ feet and we consider gravity to be $g = 10$ ft/s$^2$, find the exact value of the period $T$. Rationalize the denominator and extract any possible factors.
Calculate the exact length of the main diagonal $D$ of a cube whose edge measures $a = \sqrt{5}$ in. Remember that the diagonal of a cube is calculated as…
Calculate the exact length of the main diagonal $D$ of a cube whose edge measures $a = \sqrt{5}$ in. Remember that the diagonal of a cube is calculated as $D = \sqrt{a^2 + a^2 + a^2}$. Express the result in its simplest possible form.
The facade of a cabin has the shape of an isosceles triangle. The base of the triangle measures $\sqrt{6}$ feet and the two equal sides measure $\sqrt{15}$ feet each.…
The facade of a cabin has the shape of an isosceles triangle. The base of the triangle measures $\sqrt{6}$ feet and the two equal sides measure $\sqrt{15}$ feet each. Calculate the exact value of the height $h$ above the base and the total area of the facade. Rationalize the results if roots appear in the denominator.
Back to the start: $\sqrt{72} = \sqrt{36\cdot2} = 6\sqrt{2} \approx 8.485$, so your calculator and the answer key agree. And with the conjugate, $\dfrac{\sqrt{2}}{\sqrt{3}+\sqrt{2}} = \dfrac{\sqrt{2}(\sqrt{3}-\sqrt{2})}{3-2} = \sqrt{6}-2$.