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For example: An exponent of 1/3 = Do a cube root. If you convert it to decimal form: 1/3 = 0.33333... with the 3 repeating. If it gets rounded to 0.3, the exponent would then be 3/10 which means do the 10th root, then cube the result.
- Rational exponents and radicals | College Algebra | Math - Khan Academy
Unit 13: Rational exponents and radicals. Did you know that...
- Rational exponents and radicals | College Algebra | Math - Khan Academy
Rational exponents combine powers and roots of the base, and negative exponents indicate that the reciprocal of the base is to be used. Recall that a rational number is one that can be expressed as a ratio of integers, like 21/4 or 2/5. Irrational numbers like π or e cannot be so represented.
We previously learned about integer powers—first positive and then also negative. But what does it mean to raise a number to the 2.5 power? In Algebra 2, we extend previous concepts to include rational powers. We'll define how they work, and use them to rewrite exponential expressions in various ways.
Example \(\PageIndex{7}\): Solve an Equation involving Rational Exponents and Factoring. Solve \(3x^{\tfrac{3}{4}} = x^{\tfrac{1}{2}}\). Solution. This equation involves rational exponents as well as factoring rational exponents. Let us take this one step at a time.
Unit 13: Rational exponents and radicals. Did you know that square and cube roots are a kind of exponent? They're just so common, they get a special symbol. Now, we'll be able to use the exponent properties to simplify expressions whether they have exponent or radical symbols.
In this section, we will define what rational (or fractional) exponents mean and how to work with them. All of the rules for exponents developed up to this point apply. In particular, recall the product rule for exponents. Given any rational numbers m and n, then.
A rational exponent is written as \(x^{ \frac{a}{b}},\) with base \(x\) and exponent \( \frac{a}{b}\). This expression is equivalent to the \( b^{\text{th}}\)-root of \(x \) raised to the \(a^{\text{th}}\)-power, or \( \sqrt[b]{x^a}\). For example, \(8^{\frac{2}{3}}\)could be rewritten as \(\sqrt[3]{8^2}.\)
