Roots

We are asked to determine the index and radicand of the given radical expression. Let's recall how to interpret a radical expression. sqrt(a) A radical sign is used to represent the {\color{#FD9000}{n^\text{th}}} root of a number a. The number under the radical sign is called the radicand and the number written in the upper left of the radical sign is called the index. With this in mind, let's now consider the given number. sqrt(32) The number written under the radical sign is 32, so 32 is radicand. The number written in the upper left of the radical sign is 5, which means that the index is 5.

Let's consider the given expression like we did in Part A. sqrt(-119) The number written in the upper left of the radical sign is 7, which represents the index. The number written under the radical sign is -119, which represents the radicand.

We want to find the square root of 169. sqrt(169) We need to think of a number that, when raised to the second power, results in 169. This number is 13. 169 &= 13^2 &⇓ sqrt(169) &= 13 Therefore, the positive square root of 169 is 13.

This time we will find the square root of a fraction. sqrt(81/144) Notice that 81 is the square of 9 and 144 is the square of 12. With this in mind, we will now use the inverse of the Power of a Quotient Property.

We ended with the square of a fraction in the radicand. According to the definition of the square root, we can say that the square root of 81144 is 912. 81/144 = (9/12)^2 ⇒ sqrt(81/144)=9/12 Notice that we can also reduce the resulting fraction by 3!

The square root of 81144 can be simplified to 34.

This time we will find the square root of a decimal number. Let's start by rewriting the decimal number as a fraction. 2.56=256/100 Notice that both the numerator and denominator are perfect squares. We know that 256 is the square of 16 and 100 is the square of 10. Let's now rewrite these numbers as powers.

We ended with the square of a decimal number. 2.56 &= 1.6^2 &⇕ sqrt(2.56) &= 1.6 We found that the square root of 2.56 is equal to 1.6.

We want to find the cube root of 64. We need to think of a number that results in 64 when multiplied by itself three times. ^3 =64 Since 4 raised to the third power is 64, the number we are looking for is 4. 64 &= 4^3 &⇓ sqrt(64) &= 4

Now we want to look for a number that, when multiplied by itself three times, results in -216. Notice that the radicand is negative and the index 3 is odd. This means that the result will be negative too. ^3=-216 We know that -6 raised to the third power is -216. Then, the cube root of -216 is -6. -216 &= (-6)^3 &⇓ sqrt(-216) &= -6

We want to find the cube root of a fraction. Let's start by thinking of numbers that give us the values in the numerator and denominator when multiplied by themselves three times. ^3=343 ^3=512 Notice that these numbers are perfect cubes because 343 is the third power of 7 and 512 is the third power of 8. 7^3=343 8^3=512 Let's now use these numbers to rewrite the fraction in the radicand.

Since 78 multiplied by itself three times gives 343512, we can say that the cube root of 343512 is 78. 343/512=(7/8)^3 ⇒ sqrt(343/512)=7/8

This time we will look for a number that, when multiplied by itself three times, results in -0.125. Notice that the radicand is a negative decimal number. ^3=-0.125 Resulting a negative number when the number is multiplied by itself three times means that the number itself is also negative. Let's rewrite the decimal number in the radicand as a fraction. -0.125=- 125/100 Great! We will now consider the numerator and denominator separately like we did in Part A.

We found that -0.5 multiplied by itself three times gives -0.125. This means that the cube root of -0.125 is -0.5. -0.125= (-0.5)^3 ⇒ sqrt(-0.125)=-0.5

We will examine the given numbers one by one to determine which are irrational numbers. Let's start by recalling some characteristics of irrational numbers.

• Irrational numbers cannot be written as a ratio of two integers.
• A number is irrational if it is not rational.
• The decimal expansion of irrational numbers is not repeating and non-terminating.
• The square root of any whole number that is not a perfect square is irrational.
• The cube root of any integer that is not a perfect cube is irrational.

Let's start with the famous number π and take a look at its decimal expansion. π=3.14159265359 ... ✓ Since it is not repeating and non-terminating, we can say that it is irrational. Next, let's have a look at sqrt(16). Note that 16 is a perfect square. 16 = 4^2 ⇒ sqrt(16) = 4 * Since 4 is a rational number, we have that sqrt(16) is not irrational. Now let's consider sqrt(216). Note that 216 is a perfect cube. 216 = 6^3 ⇒ sqrt(216) = 6 * Once again we ended with a rational number, which means that sqrt(216) is not irrational. Let's now look at the decimal number 0.32. It is a rational number because it is a terminating decimal. We can also express it as a ratio. 0.32 = 32/100 * Then, we will consider sqrt(32). Notice that 32 is not a perfect square, meaning that we cannot remove the square root sign. As a result, sqrt(32) is irrational. sqrt(32) ✓ Moving on to sqrt(12), note that 12 is not a perfect cube. This means that sqrt(12) is also irrational. sqrt(12) ✓ Last is -sqrt(12). Notice that 12 cannot be rewritten as a number squared, so we cannot remove the square root sign. This means that -sqrt(12) is irrational. -sqrt(1/2) ✓ Finally, let's make a table to see which numbers are irrational and which numbers are rational.