nth root. Rational Exponents.

The nth degree root ☰

The nth degree root of a number can be found using the following formula: \( \sqrt[n]{a} = a^\frac{1}{n} \) Where "\(a\)" is the number to be rooted, and "\(n\)" is the degree of the root.
For instance, the fourth degree root of 81 can be found using the formula above:
\( \sqrt[4]{81}=81^\frac{1}{4} \)
We can simplify the expression above by using the fact that 81 is equal to 3 raised to the fourth power:
\( \sqrt[4]{81}=(3^4)^\frac{1}{4}=3^{4\cdot \frac{1}{4}}=3^1=3 \)
Therefore, the fourth degree root of 81 is 3.

It's worth noting that some nth degree roots can be irrational, meaning they cannot be expressed as a ratio of two integers. For example, the square root of \(2 (\sqrt{2}) \) is an irrational number because it cannot be written as a fraction.

Properties of nth Roots:

• Product Property: The nth root of a product is equal to the product of the nth roots of the factors. That is, for any non-negative real numbers \(a\) and \(b\) and any positive integer \(n\), \( \sqrt[n]{ab}=\sqrt[n]{a}\cdot \sqrt[n]{b} \)
  For example, \( \sqrt{4\cdot 9}=\sqrt{4}\cdot \sqrt{9}=2\cdot 3=6 \).
• Quotient Property: The nth root of a quotient is equal to the quotient of the nth roots of the numerator and denominator. That is, for any non-negative real numbers \(a\) and \(b\) where \(b \neq 0 \) and any positive integer \(n\), \( \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}} \)
  For example, \( \sqrt{\frac{9}{4}}=\frac{\sqrt{9}}{\sqrt{4}}=\frac{3}{2} \)
• Power Property: The nth root of a power is equal to the power of the nth root. That is, for any non-negative real number \(a\) and any positive integers \(m\) and \(n\), \( \sqrt[n]{a^m}=a^\frac{m}{n}\)
  For example, \( \sqrt[3]{8^2}=8^\frac{2}{3}=4\)
• Radicand Property: If \(n\) is odd, then every non-negative real number has a unique nth root. If \(n\) is even, then the nth root of a non-negative real number is defined only for non-negative radicands.
  For example, \( \sqrt[3]{-8} = -2 \), but \( \sqrt{16} = 4 \) and \( \sqrt{-16} \) is not defined among the real numbers.
• Exponentiation: Exponentiation of the nth degree root is a property that tells us how to raise an nth root to a power. Specifically, for any positive integer \(n\), any non-negative real numbers \(a\) and \(b\), and any integer \(m\), \( (\sqrt[n]{a})^m = \sqrt[n]{a^m} \)
  This property means that we can simplify expressions like \( (\sqrt[3]{2})^2 \) by first raising the nth root to the power, and then taking the nth root of the result: \( (\sqrt[3]{2})^2 = \sqrt[3]{2^2}=\sqrt[3]{4} \)
  
  We can also use this property to simplify more complicated expressions involving radicals. For example, we can simplify the expression \( \sqrt[3]{2^5 \cdot 3^2} \) as follows: \( \small \sqrt[3]{2^5 \cdot 3^2}=\sqrt[3]{2^3 \cdot 2^2 \cdot 3^2}=\sqrt[3]{8\cdot 9\cdot 4}=\sqrt[3]{288} \)
• The root of the root of the nth degree: The root of the nth degree root is not a commonly used property or formula. However, one interpretation of this phrase could be the following:
  For any positive integers \(m\) and \(n\) and any non-negative real number \(a\), we have: \( \sqrt[m]{ \sqrt[n]{a} }=\sqrt[mn]{a} \)
  This property tells us that taking the mth root of the nth root of a non-negative real number \(a\) is equivalent to taking the mnth root of \(a\). For example, we have: \( \sqrt[2]{ \sqrt[3]{8} }=\sqrt[2\cdot 3]{2^3}=\sqrt{2} \)
  
  Note that this property only holds when a is non-negative, since the nth root of a negative number is not well-defined for even values of \(n\). Additionally, while this property can be useful for simplifying certain expressions involving radicals, it is not as widely applicable as some of the other properties and formulas discussed earlier.

These properties and formulas of nth root are useful for simplifying and solving problems involving radicals.

Rational exponents ☰

Rational exponents, also known as fractional exponents, are a way of representing powers and roots of a number in a more general and flexible way than using only integer exponents. A rational exponent is a number that can be expressed as a fraction, where the numerator represents the power to which the base is raised, and the denominator represents the root that is taken.

For example, let \(a\) be a positive real number and \(m\) and \(n\) be positive integers. Then the following are some examples of rational exponents:

\( a^\frac{1}{2}=\sqrt{a} \)

\( a^\frac{2}{3}=\sqrt[3]{a^2} \)

\( a^\frac{3}{4}=\sqrt[4]{a^3} \)

\( a^\frac{5}{2}=\sqrt{a^5} \)

In general, we can define a rational exponent as follows: \( a^\frac{m}{n} = \sqrt[n]{a^m} \), where \(a\) is a positive real number, \(m\) is an integer, and \(n\) is a positive integer.

Properties of exponents:

• Product rule: \(a^m \cdot a^n = a^{m+n}\)
  This property tells us that when we multiply two numbers with the same base, we can add their exponents to get the exponent of the product.
  For example, \(2^3 \cdot 2^4 = 2^{3+4}=2^7=128 \)
• Quotient rule: \(\frac{a^m}{a^n} = a^{m-n} \)
  This property tells us that when we divide two numbers with the same base, we can subtract their exponents to get the exponent of the quotient.
  For example, \(\frac{5^7}{5^3} = 5^{7-3}=625 \)
• Power rule: \( (a^m )^n=a^{m\cdot n} \)
  This property tells us that when we raise a number to a power and then raise the result to another power, we can multiply the exponents to get the exponent of the final result.
  For example, \( (2^3 )^4=2^{3\cdot 4}=2^12=4096 \)
• Negative exponent rule: \( a^{-n}=\frac{1}{a^n} \)
  This property tells us that when we have a negative exponent, we can flip the base and make the exponent positive.
  For example, \( 2^{-3}=\frac{1}{2^3}=\frac{1}{8} \)
• Zero exponent rule: \( a^0=1\)
  This property tells us that any number raised to the power of zero equals one.
  For example, \( 2^0=1 \)
• Fractional exponent rule: \(a^\frac{m}{n}=\sqrt[n]{a^m} \)
  This property tells us that when we have a fractional exponent, we can take the nth root of the base raised to the power of the numerator of the fraction.
  For example, \(2^\frac{3}{2}=\sqrt[2]{2^3}=\sqrt{8}=2\sqrt{2} \)
• Product to power rule: \((ab)^n=a^n \cdot b^n \)
  This property tells us that when we raise a product of two numbers to a power, we can distribute the power to each factor.
  For example, \( (2 \cdot 3)^4 = 2^4 \cdot 3^4=16 \cdot 81=1296 \)
• Power of a quotient rule: \( (\frac{a}{b})^n=\frac{a^n}{b^n} \)
  This property tells us that when we raise a quotient of two numbers to a power, we can distribute the power to the numerator and denominator separately.
  For example, \( (\frac{3}{2})^4=\frac{3^4}{2^4} =\frac{81}{16} \)
• Negative base rule: \( (-a)^n=(-1)^n \cdot a^n \)
  This property tells us that when we raise a negative number to an even power, the result is positive, while if we raise it to an odd power, the result is negative.
  For example, \( (-2)^4=(-1)^4 \cdot 2^4=16 \) , while \( (-2)^3=(-1)^3 \cdot 2^3=-8 \)

These properties of exponents allow us to simplify complex expressions and perform operations on them more easily. They are important not only in algebra, but also in many other fields of mathematics and science.