Just like any other number we’ve worked with, irrational numbers can be added or subtracted. When working with a calculator, enter the operation and a decimal representation will be given.
However, there are times when two irrational numbers may be added or subtracted without the calculator. This can happen only when the irrational parts of the irrational numbers are the same.
How to: Add and Subtract Irrational Numbers with the Same Irrational Part
To add or subtract two irrational numbers that have the same irrational part, add or subtract the rational parts of the numbers, and then multiply that by the common irrational part.
Let our first irrational number be [latex]a×x[/latex], where [latex]a[/latex] is the rational and [latex]x[/latex] the irrational parts.
Let our second irrational number be [latex]b×x[/latex], where [latex]b[/latex] is the rational and [latex]x[/latex] the irrational parts.
Then [latex]a×x±b×x=(a±b)×x[/latex].
Subtract the following irrational numbers.
[latex]3\sqrt{7}–8\sqrt{7}[/latex]
Since these two irrational numbers have the same irrational part, [latex]\sqrt{7}[/latex], we can subtract without using a calculator. The rational part of the first number is [latex]3[/latex]. The rational part of the second number is [latex]8[/latex]. Using the formula yields:
Since these two irrational numbers have the same irrational part, [latex]π[/latex], the addition can be performed without using a calculator. The rational part of the first number is [latex]35[/latex]. The rational part of the second number is [latex]17[/latex]. Using the formula yields
Just like any other number that we’ve worked with, irrational numbers can be multiplied or divided. When working with a calculator, enter the operation and a decimal representation will be given. Sometimes, though, you may want to retain the form of the irrational number as a rational part times an irrational part.
The process is similar to adding and subtracting irrational numbers when they are in this form. We do not need the irrational parts to match. Even though they need not match, they do need to be similar, such as both irrational parts are square roots, or both irrational parts are multiples of pi. Also, if the irrational parts are square roots, we may need to reduce the resulting square root to lowest terms.
When multiplying two square roots, use the following formula.
For any two positive numbers [latex]a[/latex] and [latex]b[/latex], [latex]\sqrt{a \times {b}}=\sqrt{a} \times \sqrt{b}[/latex]
When dividing two square roots, use the following formula.
For any two positive numbers [latex]a[/latex] and [latex]b[/latex], with [latex]b[/latex] not equal to [latex]0[/latex], [latex]\sqrt{a} \div \sqrt{b} = \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}[/latex]
How To: Multiply or Divide Irrational Numbers With Similar Irrational Parts
Step 1: Multiply or divide the rational parts.
Step 2: If necessary, reduce the result of Step 1 to lowest terms. This becomes the rational part of the answer.
Step 3: Multiply or divide the irrational parts.
Step 4: If necessary, reduce the result from Step 3 to lowest terms. This becomes the irrational part of the answer.
Step 5: The result is the product of the rational and irrational parts.
Perform the following operations without a calculator. Simplify if possible.
[latex](19\sqrt{3})×(5.6\sqrt{12})[/latex]
[latex]13 \pi \times 8 \pi[/latex]
In this multiplication problem, [latex](19\sqrt{3})×(5.6\sqrt{12})[/latex], 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. [latex]19×5.6=106.4[/latex]
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 4: If necessary, reduce the result from Step 3 to lowest terms. The radicand is [latex]36[/latex], which is the square of [latex]6[/latex]. The irrational part reduces to [latex]\sqrt{36}=6[/latex].
Step 5: The result is the product of the rational and irrational parts, which is [latex]106.4×6=638.4[/latex]
In this multiplication problem, [latex]13 \pi \times 8 \pi[/latex], notice that the irrational parts of these numbers are the same, [latex]π[/latex]. Follow the process above.
Step 1: Multiply the rational parts. [latex]13×8=104[/latex]
Step 2: If necessary, reduce the result of Step 1 to lowest terms. That result is an integer.
Step 3: Multiply the irrational parts. [latex]π×π=π^2[/latex]
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 [latex]104π^2[/latex].
Perform the following operations without a calculator. Simplify if possible.
[latex]3\sqrt{15} \div (8\sqrt{3})[/latex]
[latex]14.7\sqrt{135} \div (3\sqrt{5})[/latex]
In this division problem, [latex]3\sqrt{15} \div (8\sqrt{3})[/latex], 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. [latex]3÷8=\frac{3}{8}[/latex]
Step 2: If necessary, reduce the result of Step 1 to lowest terms. The [latex]3[/latex] and [latex]8[/latex] have no common factors, so [latex]\frac{3}{8}[/latex] is already in lowest terms.
Step 4: If necessary, reduce the result from Step 3 to lowest terms. The radicand can be reduced, which yields [latex]\sqrt{5}[/latex].
Step 5: The result is the product of the rational and irrational parts, which is [latex]\frac{3}{8}\sqrt{5}[/latex].
In this division problem, [latex]14.7\sqrt{135} \div (3\sqrt{5})[/latex], 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. [latex]14.7÷3=4.9[/latex]
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 4: If necessary, reduce the result from Step 3 to lowest terms. The radicand can be reduced, which yields [latex]\sqrt{\frac{135}{5}} = \sqrt{27} = \sqrt{ 9 \times 3}= 3\sqrt{3}[/latex]
Step 5: The result is the product of the rational and irrational parts, which is [latex]4.9×3\sqrt{3}=14.7\sqrt{3}[/latex]