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. 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: Simplify square roots 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 4: Write [latex]\sqrt{n}[/latex] in its simplified form, [latex]a\sqrt{b}[/latex].
To simplify the given radical expressions, we’ll break down the numbers into their prime factors and simplify the radicals accordingly, while also considering the powers of the variables. Here are the steps:First, factor [latex]300[/latex] into its prime factors:
[latex]300 = 2^2 \cdot 3 \cdot 5^2[/latex]
Now, extract the square roots of the perfect squares:
[latex]\begin{align} \sqrt{300} &= \sqrt{2^2 \cdot 3 \cdot 5^2} && \text{Factor the number into prime factors.} \\ &= \sqrt{2^2} \cdot \sqrt{3} \cdot \sqrt{5^2} && \text{Separate each factor under its own square root.} \\ &= 2 \cdot \sqrt{3} \cdot 5 && \text{Simplify the square roots of perfect squares.} \\ &= 10\sqrt{3} && \text{Multiply the results to get the simplified form.} \end{align}[/latex]
Simplify [latex]\sqrt{162{a}^{5}{b}^{4}}[/latex].
\begin{align} \sqrt{162a^5b^4} &= \sqrt{2 \cdot 3^4 \cdot a^5 \cdot b^4} && \text{Factor the number into prime factors and express variables.} \\ &= \sqrt{2 \cdot (3^2)^2 \cdot a^4 \cdot a \cdot (b^2)^2} && \text{Break down the expression to show squares for clarity.} \\ &= \sqrt{2} \cdot \sqrt{(3^2)^2} \cdot \sqrt{a^4} \cdot \sqrt{a} \cdot \sqrt{(b^2)^2} && \text{Separate each factor under its own square root.} \\ &= \sqrt{2} \cdot 3^2 \cdot a^2 \cdot \sqrt{a} \cdot b^2 && \text{Simplify the square roots of perfect squares.} \\ &= 9a^2b^2 \cdot \sqrt{2a} && \text{Combine the constants and simplify further to finalize.} \end{align}
For the variable [latex]x[/latex], [latex]\sqrt{x^2} = |x|[/latex] , but why is that?When you square any values, the result is always non-negative, meaning it’s either positive or zero. Then, when you take the square root of this non-negative squared value, you get back the original number without its sign—just its size or magnitude. Thus, taking the square root of [latex]x^2[/latex] always yields the absolute value of [latex]x[/latex] ensuring that we consider [latex]x[/latex] in its non-negative form.
Given the product of multiple radical expressions, we can use the product rule to combine them into one radical expression and then simplify as we did above.
Simplify the following radical expression.
[latex]\sqrt{12}\cdot \sqrt{3}[/latex]
[latex]\sqrt{50x}\cdot \sqrt{2x}[/latex]
[latex]\begin{align}\sqrt{12}\cdot \sqrt{3} & = \sqrt{12\cdot 3} && \text{Express the product as a single radical expression}. \\ &= \sqrt{36} && \text{Simplify}. \\ &= 6 \end{align}[/latex]
[latex]\begin{align*} \sqrt{50x} \cdot \sqrt{2x} &= \sqrt{50x \cdot 2x} & \text{Multiply under the radicals} \\ &= \sqrt{100x^2} & \text{Simplify the product inside the radical} \\ &= 10|x| & \text{Take the square root of \(100\) and \(x^2\)} \end{align*}[/latex]
definition
A prime factor is a factor of a number that is a prime number itself.
Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves.