Ideas
- Simplifying square and cube roots
Simplifying square and cube roots
Radical expressions have many parts as shown in the following diagram:
Index
The number on a radical symbol that indicates which type of root it represents. For instance, the index on a cube root is 3. The index on a square root is usually not written, but would be 2
Radical
A mathematical expression that uses a root, such as a square root \sqrt{\quad}, or nth root \sqrt[n]{\quad}
Radicand
The value or expression inside the radical symbol
Perfect square
A number that is the result of multiplying two of the same integer
Perfect cube
A number that is the result of multiplying three of the same integer together
Radical expressions are written in simplified radical form if the radicand cannot be factored any further. For square roots, this means there are no remaining factors of the radicand that are perfect squares and for cube roots this means there are no remaining factors of the radicand that are perfect cubes. If an expression is in simplified radical form, and there is still a number left in the radicand, the result will be irrational.
We can use the following facts to simplify radical expressions, for a, b \geq 0 and m,n positive integers,\begin{aligned} \sqrt[n]{ab} &=\sqrt[n]{a} \sqrt[n]{b} \\ \sqrt[n]{a^n}&=a \\ a^{mn}&=\left(a^m\right)^n \end{aligned}
Exploration
We can use prime factors to help us split a number into a product of a perfect square and a remainder.
If we wanted to simplify \sqrt{60}, we want to see if 60 has any factors which are perfect squares.
Using the factor tree, we can see that 60=2^2\cdot 3\cdot 5, so the perfect square factor is 2^2=4, and the remainder is 15.
\begin{aligned}\sqrt{60}&=\sqrt{2 \cdot 30} \\ &= \sqrt{2 \cdot 3\cdot 10}\\ &= \sqrt{2\cdot 2\cdot 3\cdot 5}\\ &=\sqrt{2^2}\cdot \sqrt{3\cdot 5} \\ &= 2\sqrt{15}\end{aligned}
- How can you identify if a number has a perfect square factor after drawing a factor tree?
- Is it possible to get two different factor trees for the same number?
- Is it possible to get two different prime factorizations?
- Can every radical be simplified?
- How would you rewrite 180 as a product of a perfect square and a remainder?
- What would happen if we didn't use the largest perfect square?
Prime factorization method
We can use a few steps to help us simplify any radical:
- Find prime factorization of radicand
- Group factors in groups equal to index of radical expression
- Use multiplication property of radicals
- Simplify any rational factors
Perfect square method
This is the quickest method for simplifying a radical.
- Find largest perfect square (or cube) factor of radicand
- Use multiplication property of radicals
- Use the properties of radicals to factor
- Simplify any perfect square (or cube) factors
Examples
Example 1
Simplify \sqrt{12}
Worked Solution
Create a strategy
Split 12 into its prime factors and use \sqrt{a\cdot b}=\sqrt{a}\cdot\sqrt{b}.
Apply the idea
| \displaystyle \sqrt{12} | \displaystyle = | \displaystyle \sqrt{2\cdot2\cdot3} | Find prime factorization of 12 |
| \displaystyle \text{ } | \displaystyle = | \displaystyle \sqrt{2^2\cdot3} | Rewrite using perfect square |
| \displaystyle \text{ } | \displaystyle = | \displaystyle \sqrt{2^2}\cdot\sqrt{3} | Product of radicals property |
| \displaystyle \text{ } | \displaystyle = | \displaystyle 2\sqrt{3} | Square root of a perfect square |
Example 2
Simplify \sqrt[3]{-64}
Worked Solution
Create a strategy
Split -64 into factors. Notice the negative sign.
Apply the idea
| \displaystyle \sqrt[3]{-64} | \displaystyle = | \displaystyle \sqrt[3]{-4 \cdot -4 \cdot -4} | Factor radicand |
| \displaystyle = | \displaystyle \sqrt[3]{(-4)^3} | Rewrite using perfect cube | |
| \displaystyle = | \displaystyle -4 | Cube root of perfect cube |
Reflect and check
Negative radicands can have rational cube roots. Can the same be said for square roots?
Example 3
Simplify the expression 3\sqrt{18}.
Worked Solution
Create a strategy
To simplify the expression, we can first simplify the square root and then multiply the result by the coefficient outside the root.
Apply the idea
| \displaystyle 3\sqrt{18} | \displaystyle = | \displaystyle 3\sqrt{9 \times 2} | Factor 18 into 9 \times 2 |
| \displaystyle = | \displaystyle 3 \times \sqrt{9} \times \sqrt{2} | Use the property of square roots that \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} | |
| \displaystyle = | \displaystyle 3 \times 3 \times \sqrt{2} | Find the square root of 9, which is 3 | |
| \displaystyle = | \displaystyle 9\sqrt{2} | Multiply the coefficients together |
Therefore, the simplified form of 3\sqrt{18} is 9\sqrt{2}.
Reflect and check
Let's check our solution by substituting back into the original expression and evaluating both sides.
| \displaystyle 3\sqrt{18} | \displaystyle = | \displaystyle 3 \times 4.24 | Substitute \sqrt{18} with its decimal approximation 4.24 |
| \displaystyle = | \displaystyle 12.72 | Evaluate | |
| \displaystyle 9\sqrt{2} | \displaystyle = | \displaystyle 9 \times 1.41 | Substitute \sqrt{2} with its decimal approximation 1.41 |
| \displaystyle = | \displaystyle 12.69 | Evaluate |
The values on both sides are approximately equal, confirming that our solution is correct. The slight difference is due to the rounding off of the square roots to two decimal places.
Idea summary
Radical expressions can be simplified if the radicand is divisible by any perfect squares (for square root) or perfect cubes (for cube root). If a number can be simplified to remove the radical, it is rational. Otherwise, it is irrational.
