Radicals Study Guide (page 2)
Introduction to Radicals
Math is radical!
— Bumper Sticker
In this lesson, you'll learn how to work with radicals and fractional exponents and how to solve equations with radicals.
A radical is a root of a quantity. You can think of radicals and exponents as opposites. If we raise a quantity, x, to the fourth power, we multiply it four times: x4 = (x)(x)(x)(x). If we take the fourth root of x, we are looking for the number that, when multiplied four times, is equal to x. We show the fourth root of x as 4√x. The √ symbol is called the radical symbol. The quantity under the radical symbol is called the radicand. The number just outside the radical symbol is the root that must be taken. In 4√x, x is the radicand and 4 is the root. If no number appears outside the radical symbol, then 2 is the root.
Let's look at an example with numbers: 23 = (2)(2)(2) = 8. The third root of 8 is 2, because 2 is the number that, when multiplied three times, is equal to 8: 3√8 = 2.
A radicand can have an exponent. The square root of x4, √x4, is x2, because (x2)(x2) = x4.
If the root of a radical is the same as the exponent of the base of the radicand, and the root is positive, the expression can be simplified to the base of the radicand. For example, the fifth root of x to the fifth power is x. 5√x5 = x. For positive values of x, the tenth root of x to the tenth power is x, and so on.
We cannot find even roots of negative numbers. For example, we cannot find the square root of a negative number, because no number multiplied by itself is negative. A positive number multiplied by a positive number gives you a positive product, and a negative number multiplied by a negative number gives you a positive product. We can find the odd roots of some negative numbers. For instance, 3√-x3 = –x, because (–x)(–x)(–x) = –x3.
If a coefficient appears in the radicand, we must take the root of the coefficient as well as the base. √16z = 4z4, because (4z4)(4z4) = 16z8. The square root of an even exponent is always half the exponent.
A radical itself can have a coefficient. It is written to the left of the radical symbol. 5√b6 is 5 times the square root of b6. Half of 6 is 3, so the square root of b6 is b3. Five times b3 is 5b3.
The exponent of a variable can be written as a fraction, as we saw in Lesson 8. The numerator of a fractional exponent is the power to which the variable is raised. The denominator of the fractional exponent is the root to take of the variable. is equal to 3√g7, which is the third root of g to the seventh power. g7 does not have a third root that we can easily calculate, so we cannot simplify this expression any further. We were, though, able to find the square root of b6.The reason why becomes clearer when we write the square root of b to the sixth power as a fractional exponent. Because we are raising b to the sixth power, the numerator of the fraction is 6, and since we are taking the square, or second, root of b, the denominator of the fraction is 2: .The fraction reduces to 3, just as the square root of b to the sixth power is equal to b3.
Try to memorize the squares of numbers up to 20, and their square roots. This will help you work with exponents and radicals faster.
Solving Equations with Radicals
Now that we know how to work with radicals, we can use them to help us solve equations. So far, we have used addition, subtraction, multiplication, and division to solve equations. Sometimes, we will need to raise both sides of an equation to a power, or take a root of both sides of an equation in order to find our answer.
x2 = 64
How can we find the value of x? Addition, subtraction, multiplication, and division cannot help us. However, we can get x alone on the left side of the equation if we take the square root of both sides of the equation. Why the square root? Because the exponent of x is 2. If the exponent of x was 5, we would take the fifth root of x to get x alone. Remember the tip we learned earlier: If the exponent and root of a base are the same, the term can be simplified to just the base or its absolute value.
√x2 = √64
The square root of x2 is the absolute value of x and the square root of 64 is 8, since (8)(8) = 64. We are left with x = 8. This isn't our only answer, though. It's true that the square root of 64 is 8, but there is another number that, when squared, equals 64: –8. Remember, (–8)(–8) = 64.
Our answers are x = 8, –8.
4x2 = 144
To solve for x, we start by dividing both sides of the equation by 4.
x2 = 36
Now, we can take the plus and minus square roots of 36. Because (6)(6) = 36 and (–6)(–6) = –36, our answers are x = 6, –6.
We can also use exponents to help us simplify radicals. To remove the radical symbol from an equation, raise it to an exponent that is equal to the root of the radical.
4√r = 3
Because the fourth root of r is equal to 3, raise both sides of the equation to the fourth power to remove the radical symbol.
r = 81
Find practice problems and solutions for these concepts at Radicals Practice Questions.
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