HN8 - Working with surds

Exact roots

Is $\sqrt{72}\div\sqrt2$ a whole number? Both roots look awkward as decimals, but their exact forms reveal a simple relationship.

A rational number can be written as a fraction of integers with a nonzero denominator. For example, $\sqrt{25}=5$ and $\sqrt{9/4}=3/2$ are rational. Some roots are irrational: no fraction of integers equals them exactly.

A decimal approximation such as $1.414$ is useful for checking size, but it is not equal to $\sqrt2$. Retain root notation when an exact answer is required.

To simplify a square root, split off the largest square-number factor. Since $72=36\times2$, the square factor $36$ comes out as its root, $6$.

[DIAGRAM: asset_name: A square factor inside a root; asset_slug: square-factor-surd; caption: Thirty-six equal tiles make a square with six tiles along each side.; description: One square tile has area 2 and side length square root of 2. A six-by-six arrangement of the same tiles has area 72 and side length 6 times square root of 2, showing square root of 72 equals 6 times square root of 2.]

Now put the same numerator over a square-number denominator. Simplify the two roots separately.

If you choose a smaller square factor, continue until no square factor greater than $1$ remains inside the root. For example, $\sqrt{72}=\sqrt{4\times18}=2\sqrt{18}=2\times3\sqrt2=6\sqrt2$.

Simplify another fraction using square factors.

Surd arithmetic

Like surds combine in the same way as equal units. Three lots of $\sqrt2$ plus two lots of $\sqrt2$ make five lots: $3\sqrt2+2\sqrt2=5\sqrt2$.

Simplify roots before deciding whether they are alike. For example, $\sqrt{18}+\sqrt8=3\sqrt2+2\sqrt2=5\sqrt2$. But $\sqrt2+\sqrt3$ has unlike roots and stays as a sum.

For products, multiply rational factors and root factors separately: $(3\sqrt2)(2\sqrt3)=6\sqrt6$. In particular, $\sqrt2\times\sqrt2=2$. An exact integer can emerge from a product of irrational numbers.

To multiply brackets, distribute every term in the first bracket across every term in the second. With numbers, $(3+2)(4+1)=3\times4+3\times1+2\times4+2\times1$. The same distributive law works when a term contains a surd.

Change the second plus sign to a minus sign. Two products become negative, so keep the signs attached to their terms.

Division by a rational number also acts on the coefficient: $6\sqrt2\div3=2\sqrt2$. We can now answer the opening question exactly: $\sqrt{72}\div\sqrt2=6\sqrt2\div\sqrt2=6$, since $\sqrt2\ne0$.

Use four signed products in this new calculation.

Rationalising denominators

An exact fraction may have an irrational denominator. Rationalising the denominator rewrites it with a rational denominator while keeping its value unchanged.

Multiply numerator and denominator by the same nonzero expression. This multiplies the fraction by $1$. For a denominator $\sqrt b$, multiplying by $\sqrt b$ makes the denominator $b$.

If the denominator has two terms, multiplying by itself usually leaves a surd. Instead, use the same two terms with the middle sign reversed. For example,

The root terms cancel. The paired expressions are called conjugates. Their product is rational; check that it is nonzero before using the pair as a fraction multiplier.

Add $2$ to the denominator. A conjugate now removes the root without changing the fraction's value.

A rational denominator does not require an integer answer. For example, $1/(3+\sqrt2)=(3-\sqrt2)/7$, since the conjugate product is $9-2=7$. The result may still contain a surd in its numerator.

Use a conjugate with a denominator that does not become $1$.

Quick Recap

• A surd keeps an irrational root exact; terminating decimal approximations lose that exact value.
• Extract square factors using their square roots: $\sqrt{72}=6\sqrt2$.
• Simplify before collecting like roots; multiply every signed pair in a product of brackets.
• Rationalise by multiplying numerator and denominator equally, using a conjugate for two-term denominators.

The opening quotient is the whole number $6$. Keeping the roots exact makes that clear.