3.5 Irrational Numbers

1. The prime factorization of 45 is $45=3^2\times 5$. Since the 5 is not raised to an even power, 45 is not a perfect square.
2. The prime factorization of 81 is $3^4$. All the prime factors are raised to even powers, so 81 is a perfect square.
3. We must determine if both the numerator and denominator of $\frac{9}{28}$ are perfect squares for the rational number to be a perfect square. The numerator is 9, and as mentioned above, 9 is a perfect square (it is 3 squared). Now we check the prime factorization of the denominator, 28, which is $28=2^2\times 7$. Since 7 is not raised to an even power, 28 is not a perfect square. Since the denominator is not a perfect square, $\frac{9}{28}$ is not a perfect square.
4. We must determine if both the numerator and denominator of $\frac{144}{400}$ are perfect squares for the rational number to be a perfect square. The numerator is 144. The prime factorization of 144 is $144=2^4\times 3^2$. Since all the prime factors of 144 are raised to even powers, 144 is a perfect square. Now we check the prime factorization of the denominator, 400, which is $400=2^4\times 5^2$. Since all the prime factors of 400 are raised to even powers, 400 is a perfect square. Since the numerator and denominator of $\frac{144}{400}$ are perfect squares, $\frac{144}{400}$ is a perfect square.

1. 35 can be factored as $5\times 7$, showing that 35 is not the square of an integer or a rational number. This mean its square root is an irrational number.
2. Since $0.\overline{15}$ is a decimal with a repeating pattern, it is rational, so it is not an irrational number.
3. $121={11}^2$. Since 121 is the square of an integer, its square root is a rational number.
4. Since $4\pi$ is a multiple of pi, it is irrational. In this case, the rational part of the number is 4, while the irrational part is $\pi$.

Begin by finding the largest perfect square that is a factor of 180. We can do this by writing out the factor pairs of 180:

Looking at the list of factors, the perfect squares are 4, 9, and 36. The largest is 36, so we factor the into $36\times 5=6^2\times 5$. In the formula, $a=6$ and $b=5$. Apply $\sqrt{a^2\times b}=\sqrt{a^2}\times \sqrt{b}$.

$\sqrt{6^2\times 5}=\sqrt{6^2}\times \sqrt{5}$

The simplified form of $\sqrt{180}$ is $6\sqrt{5}$. In this example, the 6 is the rational part, and the $\sqrt{5}$ is the irrational part.

Begin by finding the largest perfect square that is a factor of 330. We can do this by writing out the factor pairs of 330:

Looking at the list of factors, there are no perfect squares other than 1, which means $\sqrt{330}$ is already expressed in lowest terms. In this case, 1 is the rational part, and $\sqrt{330}$ is the irrational part. Though we could write this as $1\sqrt{330}$, but the product of 1 and any other number is just the number.

Begin by finding the largest perfect square that is a factor of 2,548. We can do this by writing out the factor pairs of 2,548:

Looking at the list of factors, the perfect squares are 4, 49, and 196. The largest is 196, so we factor the 2,548 into $196\times 13={14}^2\times 13$. In the formula, $a=14$ and $b=5$. Apply $\sqrt{a^2\times b}=\sqrt{a^2}\times \sqrt{b}$.

$\sqrt{{14}^2\times 13}=\sqrt{{14}^2}\times \sqrt{13}$

The simplified form of $\sqrt{\mathrm{2,548}}$ is $14\sqrt{13}$. In this example, 14 is the rational part, and $\sqrt{13}$ is the irrational part.

Since these two irrational numbers have the same irrational part, $\sqrt{7}$, we can subtract without using a calculator. The rational part of the first number is 3. The rational part of the second number is 8. Using the formula yields $3\sqrt{7}-8\sqrt{7}=\left(3-8\right)\times \sqrt{7}=-5\sqrt{7}$.

Since these two irrational numbers have the same irrational part, $\pi$, the addition can be performed without using a calculator. The rational part of the first number is 35. The rational part of the second number is 17. Using the formula yields $35\pi +17\pi =(35+17)\times \pi =52\pi$.

The two numbers being subtracted do not have the same irrational part, so the operation cannot be performed.

1. In this division problem, $3\sqrt{15}÷\left(8\sqrt{3}\right)$, notice that the irrational parts of these numbers are similar. They are both square roots, so follow the steps given above.

   Step 1: Divide the rational parts. $3÷8=\frac{3}{8}$

   Step 2: If necessary, reduce the result of Step 1 to lowest terms. The 3 and 8 have no common factors, so $\frac{3}{8}$ is already in lowest terms.

   Step 3: Divide the irrational parts. $\sqrt{15}÷\sqrt{3}=\frac{\sqrt{15}}{\sqrt{3}}=\sqrt{\frac{15}{3}}$

   Step 4: If necessary, reduce the result from Step 3 to lowest terms. The radicand can be reduced, which yields $\sqrt{5}$.

   Step 5: The result is the product of the rational and irrational parts, which is $\frac{3}{8}\sqrt{5}$.

2. In this division problem, $14.7\sqrt{135}÷\left(3\sqrt{5}\right)$, notice that the irrational parts of these numbers are similar. They are both square roots, so follow the steps given above.

   Step 1: Divide the rational parts. $14.7÷3=4.9$

   Step 2: If necessary, reduce the result of Step 1 to lowest terms. This rational number is expressed as a decimal so will not be reduced.

   Step 3: Divide the irrational parts. $\sqrt{135}÷\sqrt{5}=\frac{\sqrt{135}}{\sqrt{5}}=\sqrt{\frac{135}{5}}$

   Step 4: If necessary, reduce the result from Step 3 to lowest terms. The radicand can be reduced, which yields $\sqrt{\frac{135}{5}}=\sqrt{27}=\sqrt{9\times 3}=3\sqrt{3}$.

   Step 5: The result is the product of the rational and irrational parts, which is $4.9\times 3\sqrt{3}=14.7\sqrt{3}$.

1. In this multiplication problem, $\left(19\sqrt{3}\right)\times \left(5.6\sqrt{12}\right)$, notice that the irrational parts of these numbers are similar. They are both square roots. Follow the process above.

   Step 1: Multiply the rational parts. $19\times 5.6=106.4$

   Step 2: If necessary, reduce the result of Step 1 to lowest terms. This rational number is expressed as a decimal and will not be reduced.

   Step 3: Multiply the irrational parts. $\sqrt{3}\times \sqrt{12}=\sqrt{3\times 12}=\sqrt{36}$

   Step 4: If necessary, reduce the result from Step 3 to lowest terms. The radicand is 36, which is the square of 6. The irrational part reduces to $\sqrt{36}=6$.

   Step 5: The result is the product of the rational and irrational parts, which is $106.4\times 6=638.4$.

   Notice that sometimes multiplying or dividing irrational numbers can result in a rational number.

2. In this multiplication problem, $13\pi \times 8\pi$, notice that the irrational parts of these numbers are the same, $\pi$. Follow the process above.

   Step 1: Multiply the rational parts. $13\times 8=104$

   Step 2: If necessary, reduce the result of Step 1 to lowest terms. That result is an integer.

   Step 3: Multiply the irrational parts. $\pi \times \pi ={\pi }^2$

   Step 4: If necessary, reduce the result from Step 3 to lowest terms. This cannot be reduced.

   Step 5: The result is the product of the rational and irrational parts, which is $104{\pi }^2$.

1. The square root in the denominator is $\sqrt{7}$. In order to rationalize the denominator of $\frac{5}{\sqrt{7}}$, we need to multiply the numerator and denominator by $\sqrt{7}$ and simplify.
   $\frac{5}{\sqrt{7}}=\frac{5\sqrt{7}}{\sqrt{7}\times \sqrt{7}}=\frac{5\sqrt{7}}{7}$
   The square root is in simplified form, so the final answer is $\frac{5\sqrt{7}}{7}$.
2. The square root in the denominator is $\sqrt{10}$.
   Step 1: In order to rationalize the denominator of $\frac{3\sqrt{6}}{2\sqrt{10}}$, we need to multiply the numerator and denominator by $\sqrt{10}$ and simplify.
   $\frac{3\sqrt{6}}{2\sqrt{10}}=\frac{3\sqrt{6}\times \sqrt{10}}{2\sqrt{10}\times \sqrt{10}}=\frac{3\sqrt{60}}{2\times 10}=\frac{3\sqrt{60}}{20}$

   Step 2: The 60 under the square root can be factored into the following factor pairs:

   $1\times 602\times 303\times 204\times 155\times 126\times 10$

   Step 3: The largest square factor of 60 is 4, so we simplify the $\sqrt{60}$ in the numerator into $2\sqrt{15}$. We also cancel any common factors.

   $\frac{3\sqrt{60}}{20}=\frac{3\times 2\sqrt{15}}{20}=\frac{6\sqrt{15}}{20}=\frac{3\sqrt{15}}{10}$

   This is completely simplified.

Step 1: We recognize that the denominator is the sum of two numbers where one or both involve square roots. This means the conjugate can be used to remove the square root from the denominator.

Step 2: To do so, we multiply the numerator and the denominator each by the conjugate of the denominator. Since the denominator is $6+\sqrt{10}$, the conjugate we will use is $6-\sqrt{10}$.

Step 3: The conjugate is multiplied by the numerator and the denominator.

$\frac{4}{6+\sqrt{10}}\times \frac{6-\sqrt{10}}{6-\sqrt{10}}$

Step 4: Remembering how a number times its conjugate works, this becomes

$\frac{4}{6+\sqrt{10}}\times \frac{6-\sqrt{10}}{6-\sqrt{10}}=\frac{4\times (6-\sqrt{10})}{6^2-{\left(\sqrt{10}\right)}^2}$.

Step 5: In the numerator, we apply the distributive property. Using it yields
$\frac{4\times (6-\sqrt{10})}{6^2-{\left(\sqrt{10}\right)}^2}=\frac{24-4\sqrt{10}}{36-10}=\frac{24-4\sqrt{10}}{26}$.

Step 6: Notice that the denominator no longer contains a square root. It has been rationalized. If desired, this can then be written as a rational number minus an irrational number, by recalling that $\frac{a-b}{c}=\frac{a}{c}-\frac{b}{c}$.

Applying that to the answer, we have $\frac{4}{6+\sqrt{10}}=\frac{24-4\sqrt{10}}{26}=\frac{24}{26}-\frac{4\sqrt{10}}{26}$.

Step 7: With a bit of cancellation, this reduces to $\frac{4}{6+\sqrt{10}}=\frac{24}{26}-\frac{4\sqrt{10}}{26}=\frac{12}{13}-\frac{2\sqrt{10}}{13}$.