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Suppose we know that We want to find what number raised to the 3rd power is equal to 8. Since we say that 2 is the cube root of 8.
The n th root of is a number that, when raised to the n th power, gives For example, is the 5th root of because If is a real number with at least one n th root, then the principal n th root of is the number with the same sign as that, when raised to the n th power, equals
The principal n th root of is written as where is a positive integer greater than or equal to 2. In the radical expression, is called the index of the radical.
If is a real number with at least one n th root, then the principal n th root of written as is the number with the same sign as that, when raised to the n th power, equals The index of the radical is
Simplify each of the following:
Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index is even, then cannot be negative.
We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an n th root. The numerator tells us the power and the denominator tells us the root.
All of the properties of exponents that we learned for integer exponents also hold for rational exponents.
Rational exponents are another way to express principal n th roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is
Given an expression with a rational exponent, write the expression as a radical.
Write as a radical. Simplify.
The 2 tells us the power and the 3 tells us the root.
We know that because Because the cube root is easy to find, it is easiest to find the cube root before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.
Write using a rational exponent.
The power is 2 and the root is 7, so the rational exponent will be We get Using properties of exponents, we get
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