Properties of Radicals and Rationalizing Denominators
Emma's calculator says the side of her square patio (area $50$ ft²) is $7.0710678\ldots$ ft. That is not a length she can mark on a board, and it is not exact. In this section you will learn to write $\sqrt{50}$ as a clean exact value, and to answer questions like "why is $\dfrac{1}{\sqrt5+2}$ exactly $\sqrt5-2$?"
Simplifying a Radical
A radical is in simplest form when the radicand has no perfect $n$th-power factor, there are no fractions inside the root, no radical in a denominator, and the index is as small as possible. Some problems say extract factors from a radical or introduce factors into it. That just means pulling perfect powers out of the root, or pushing a number back in.
Variables
When variables appear, assume they are positive. (For even indices you would otherwise write absolute values, such as $\sqrt{x^2}=|x|$.) Each variable comes out in groups of $n$ copies, just like numbers.
Extract all possible factors from the following radicals: a) $\sqrt{72}$ b) $\sqrt[3]{250}$
Extract all possible factors from the following radicals:
- a) $\sqrt{72}$
- b) $\sqrt[3]{250}$
Introduce the integer factors into the radical and simplify if possible: a) $3 \sqrt{5}$ b) $2 \sqrt[4]{3}$
Introduce the integer factors into the radical and simplify if possible:
- a) $3 \sqrt{5}$
- b) $2 \sqrt[4]{3}$
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:
- a) $\sqrt[6]{27}$
- b) $\sqrt[8]{16}$
Radicals Inside Radicals
Express as a single radical and simplify the result: a) $\sqrt{\sqrt[3]{64}}$ b) $\left(\sqrt[3]{\sqrt{2}}\right)^6$
Express as a single radical and simplify the result:
- a) $\sqrt{\sqrt[3]{64}}$
- b) $\left(\sqrt[3]{\sqrt{2}}\right)^6$
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:
$\sqrt{2 \cdot \sqrt[3]{2}}$
Simplify the following expression with nested radicals, expressing the result as a single root: $\sqrt{2 \cdot \sqrt[3]{4 \cdot \sqrt{8}}}$
Simplify the following expression with nested radicals, expressing the result as a single root:
$\sqrt{2 \cdot \sqrt[3]{4 \cdot \sqrt{8}}}$
Adding and Subtracting Like Radicals
Like radicals have the same index and the same radicand. Only like radicals can be combined, exactly like like terms in algebra: $3x + 5x = 8x$, so $3\sqrt2 + 5\sqrt2 = 8\sqrt2$.
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) $\frac{1}{2}\sqrt{7} + \frac{3}{2}\sqrt{7} - \sqrt{7}$
Perform the following operation, first extracting the necessary factors so that the radicals are like radicals: $3\sqrt{50} + 2\sqrt{18} - \sqrt{8}$
Perform the following operation, first extracting the necessary factors so that the radicals are like radicals:
$3\sqrt{50} + 2\sqrt{18} - \sqrt{8}$
Multiplying and Dividing Radicals
With the same index, multiply or divide the radicands. With different indices, first rewrite everything with a common index.
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:
- a) $\sqrt{2} \cdot \sqrt{8}$
- b) $\sqrt[3]{4} \cdot \sqrt[3]{10}$
Reduce to a common index and perform the following multiplication and division of radicals: $\frac{\sqrt{2} \cdot \sqrt[3]{3}}{\sqrt[6]{6}}$
Reduce to a common index and perform the following multiplication and division of radicals:
$\frac{\sqrt{2} \cdot \sqrt[3]{3}}{\sqrt[6]{6}}$
Rationalizing a Monomial Denominator
A radical in the denominator is not wrong, but it is not standard form. It is also awkward to compute with. To rationalize the denominator, multiply the top and bottom by whatever makes the denominator a rational number. This does not change the value, because you are multiplying by $1$.
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:
- a) $\frac{6}{\sqrt{3}}$
- b) $\frac{10}{\sqrt{5}}$
Rationalize the following fractions with roots of index greater than 2: a) $\frac{10}{\sqrt[5]{8}}$ b) $\frac{3}{\sqrt[4]{27}}$
Rationalize the following fractions with roots of index greater than 2:
- a) $\frac{10}{\sqrt[5]{8}}$
- b) $\frac{3}{\sqrt[4]{27}}$
Extract all possible factors from the following root and simplify the resulting expression: $\sqrt[3]{\frac{81 x^4 y^7}{16 z^5}}$
Extract all possible factors from the following root and simplify the resulting expression:
$\sqrt[3]{\frac{81 x^4 y^7}{16 z^5}}$
Special Products with Radicals
The special products you know from algebra work with radicals too. Treat $\sqrt a$ and $\sqrt b$ as the two terms. The key fact is $\left(\sqrt a\right)^2 = a$.
Prove, by expanding the special products and simplifying, that the result of the following expression is an integer: $(\sqrt{7} - \sqrt{3})^2 + (\sqrt{7} + \sqrt{3})^2$
Prove, by expanding the special products and simplifying, that the result of the following expression is an integer:
$(\sqrt{7} - \sqrt{3})^2 + (\sqrt{7} + \sqrt{3})^2$
Rationalizing with a Conjugate
If the denominator is $\sqrt a+\sqrt b$ (or $\sqrt a - 1$, and so on), multiplying by the same root will not clear it. Use the conjugate: the same two terms with the opposite sign in the middle. The conjugate of $\sqrt7+1$ is $\sqrt7-1$.
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:
- a) $\frac{4}{\sqrt{5}-1}$
- b) $\frac{2}{\sqrt{3}+1}$
Rationalize the denominator and fully simplify the following expression: $\frac{\sqrt{2}}{\sqrt{3}+\sqrt{2}}$
Rationalize the denominator and fully simplify the following expression:
$\frac{\sqrt{2}}{\sqrt{3}+\sqrt{2}}$
Combining Rationalized Fractions
Sometimes you must rationalize more than one fraction, or simplify a numerator after multiplying. Work in a fixed order.
Rationalize the denominator of the following fraction and fully simplify by operating on the numerator: $\frac{\sqrt{6}-\sqrt{2}}{\sqrt{6}+\sqrt{2}}$
Rationalize the denominator of the following fraction and fully simplify by operating on the numerator:
$\frac{\sqrt{6}-\sqrt{2}}{\sqrt{6}+\sqrt{2}}$
Perform the following combined operation with irrational fractions. To do this, first rationalize each fraction separately and then carry out the subtraction:…
Perform the following combined operation with irrational fractions. To do this, first rationalize each fraction separately and then carry out the subtraction:
$\frac{2}{\sqrt{5}-\sqrt{3}} - \frac{2}{\sqrt{5}+\sqrt{3}}$
Applications: Geometry
Exact lengths often come from the Pythagorean Theorem, and the answers are radicals. Keep them exact, simplify, and only use a calculator at the very end.
In a carpentry workshop, a rectangular board is to be cut with exact dimensions of $\sqrt{12}$ in in width and $\sqrt{27}$ in in length. Calculate the area and the…
In a carpentry workshop, a rectangular board is to be cut with exact dimensions of $\sqrt{12}$ in in width and $\sqrt{27}$ in in length. Calculate the area and the perimeter of the board, expressing the results exactly using simplified radicals.
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.
Applications: Formulas with Radicals
Many formulas in physics and design contain a square root: the period of a pendulum, speed from kinetic energy, the golden ratio. To get an exact answer, simplify the fraction, rationalize, and pull out perfect squares.
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.
In graphic design, proportions based on the golden ratio $\phi = \frac{1+\sqrt{5}}{2}$ are considered aesthetically perfect. A canvas has the shape of a golden rectangle…
In graphic design, proportions based on the golden ratio $\phi = \frac{1+\sqrt{5}}{2}$ are considered aesthetically perfect. A canvas has the shape of a golden rectangle (the quotient between its longer side and its shorter side is $\phi$). If the shorter side of the canvas measures exactly $\sqrt{5}-1$ feet, calculate the exact length of the longer side. Fully simplify the result.
In physics, the speed $v$ of an object can be found from its kinetic energy $E_c$ and its mass $m$ using the formula $v = \sqrt{\frac{2E_c}{m}}$. If a prototype vehicle…
In physics, the speed $v$ of an object can be found from its kinetic energy $E_c$ and its mass $m$ using the formula $v = \sqrt{\frac{2E_c}{m}}$. If a prototype vehicle has a mass of $m = \sqrt{3}$ kg and its kinetic energy is $E_c = 45$ J, calculate the exact value of its speed. Rationalize and extract any possible factors from the resulting root.
Back to the start: Emma's patio has side $\sqrt{50} = \sqrt{25\cdot 2} = 5\sqrt2$ ft, an exact length. And $\dfrac{1}{\sqrt5+2} = \dfrac{\sqrt5-2}{(\sqrt5+2)(\sqrt5-2)} = \dfrac{\sqrt5-2}{5-4} = \sqrt5-2$.