Simplifying Square Roots and Expressing Them in Lowest Terms
To simplify a square root means that we rewrite the square root as a rational number times the square root of a number that has no perfect square factors. The act of changing a square root into such a form is simplifying the square root.
The number inside the square root symbol is referred to as the radicand. So in the expression [latex]\sqrt{a}[/latex] the number [latex]a[/latex] is referred to as the radicand.
Before discussing how to simplify a square root, we need to introduce a rule about square roots.
the product rule for square roots
The square root of a product of numbers equals the product of the square roots of those number.
Given that [latex]a[/latex] and [latex]b[/latex] are nonnegative real numbers,
Using this formula, we can factor an integer inside a square root into a perfect square times another integer. Then the square root can be applied to the perfect square, leaving an integer times the square root of another integer. If the number remaining under the square root has no perfect square factors, then we’ve simplified the irrational number into lowest terms.
A perfect square is an integer that can be expressed as the square of another integer. For example, [latex]16[/latex], [latex]25[/latex], and [latex]36[/latex] are perfect squares because they are [latex]4^2[/latex], [latex]5^2[/latex], and [latex]6^2[/latex], respectively.
How to: To simplify the irrational number into lowest terms when [latex]n[/latex] is an integer
- Step 1: Determine the largest perfect square factor of [latex]n[/latex], which we denote [latex]a^2[/latex].
- Step 2: Factor [latex]n[/latex] into [latex]a^2×b[/latex].
- Step 3: Apply [latex]\sqrt{a^2 \times b} =\sqrt{a^2} \times \sqrt{b}[/latex].
- Step 4: Write [latex]\sqrt{n}[/latex] in its simplified form, [latex]a\sqrt{b}[/latex].
When a square root has been simplified in this manner, [latex]a[/latex] is referred to as the rational part of the number, and [latex]\sqrt{b}[/latex] is referred to as the irrational part.
Simplify the irrational number [latex]\sqrt{180}[/latex] and express in lowest terms. Identify the rational and irrational parts.
Simplify the irrational number [latex]\sqrt{330}[/latex] and express in lowest terms. Identify the rational and irrational parts.