Radicals and Rational Exponents
When you square a number (multiply it by itself) and then take its square root, you get the original number back.
\[4^2 = 16 \quad\Longrightarrow\quad \sqrt{16} = 4\]The square root is like the opposite action of squaring — just like subtraction is the opposite of addition. To “undo” squaring, we take the square root.
- \(\sqrt{}\) called the radical.
- \(a\) called the radicand (the number under the radical).
- \(\sqrt{a}\) called a radical expression.
If \(a\) is a positive number, the square root of \(a\) is a number that, when multiplied by itself, equals \(a\).
\[\sqrt{a} \times \sqrt{a} = a\]It can be positive or negative, because: \(4 \times 4 = 16\) and \((-4) \times (-4) = 16\).
However, in most cases, we use the principal square root, which means the nonnegative root. This is what you see on a calculator.
Principal Square Root
The principal square root of \(a\) is the nonnegative number that, when multiplied by itself, equals \(a\). It is written as a radical expression, with a symbol called a radical over the term called the radicand: \(\sqrt{a}\).
The Product Rule
If \(a\) and \(b\) are nonnegative, the square root of the product \(ab\) is equal to the product of the square roots of \(a\) and \(b\). \(\sqrt{ab}=\sqrt{a} \cdot \sqrt{b}\)
The Quotient Rule
The square root of the quotient \(\frac{a}{b}\) is equal to the quotient of the square roots of \(a\) and \(b\), where \(b\neq 0\).
\[\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\]Adding and Subtracting Square Roots
We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of \(\sqrt{2}\) and \(3\sqrt{2}\) is \(4\sqrt{2}\). However, it is often possible to simplify radical expressions, and that may change the radicand.
Example:
\[\sqrt{2} + \sqrt{18} = \sqrt{2} + \sqrt{9 \times 2} = \sqrt{2} + \sqrt{9} \times \sqrt{2} = \sqrt{2} + 3 \times \sqrt{2} = 4\sqrt{2}\]How to Remove Square Roots from Denominators
When we simplify a fraction in math, we try to avoid having a square root (radical) in the denominator. The process of removing the radical is called rationalizing the denominator.
We use the fact that multiplying any expression by \(1\) does not change its value. To remove the radical, we multiply both the numerator (top) and denominator (bottom) by a special form of \(1\) that makes the denominator a whole number or at least removes the radical.
Denominator with a Single Radical Term
If the denominator has only one term with a radical, multiply both numerator and denominator by that radical.
Example:
\[\frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}\]Denominator with Two Terms (Rational + Radical)
If the denominator has two terms—a rational number and a radical—multiply by the conjugate of the denominator. The conjugate is made by changing the sign between the two terms.
Example:
\[\frac{4}{2 + \sqrt{5}} \times \frac{2-\sqrt{5}}{2-\sqrt{5}} = \frac{4(2-\sqrt{5})}{(2+\sqrt{5})(2-\sqrt{5})}\]When you multiply \((2+\sqrt{5})(2-\sqrt{5})\) , you are using the difference of squares formula:
\[(a+b)(a-b)=a^2-b^2\]Here:
- \(a = 2\).
- \(b=\sqrt{5}\).
So:
\[(2+\sqrt{5})(2-\sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4-5=-1\]The denominator becomes \(−1\), so:
\[\frac{4(2-\sqrt{5})}{-1} = -8 + 4\sqrt{5}\]Understanding nth Roots
When we talk about roots in math, we often think of square roots. But there are also cube roots, 4th roots, 5th roots, and so on.
Roots are the inverse (opposite) of powers. Just like the square root is the opposite of squaring a number, the cube root is the opposite of cubing a number, and the nth root is the opposite of raising a number to the power of \(n\).
These functions are useful when we want to find the number that, when raised to a certain power, gives a specific result.
Example: Cube Roots
- If we know that: \(a^3=8\).
- we ask: What number, when raised to the 3rd power, equals \(8\)?
- Since: \(2^3=8\) we say \(2\) is the cube root of \(8\).
The nth root of a number \(a\) is a number that, when raised to the power of \(n\), equals \(a\).
Example: \((-3)^5=-243\). So \(-3\) is the 5th root of \(−243\).
The Principal nth Root
If \(a\) is a real number with at least one nth root, then the principal nth root of \(a\) is the number with the same sign as \(a\) that, when raised to the nth power, equals \(a\).
We write the principal nth root as:
\[\sqrt[\leftroot{-2} \uproot{2} n]{a}\]where:
- \(n\) is a positive integer (\(\geq 2\))
- \(n\) is called the index of the radical
- \(a\) is called the radicand
Example: \(\sqrt[\leftroot{-2} \uproot{2} 4]{81}=3\) because \(3^4=81\)