E1.17 - Simplifying surds (Extended)

E1.17 - Simplifying surds (Extended)

Can a square with area 48 have an exact side length that looks simpler than 48\sqrt{48}? Surds keep irrational values exact rather than replacing them with rounded decimals. Develop the simplifying, adding and multiplying skills in E1.17 by recognising square factors and keeping track of like terms.

Square factors

Sixteen small squares, each of area 3, can form a 44 by 44 square of area 48. Each small side is 3\sqrt3, so the large side is 434\sqrt3. This gives a picture of why 48=43\sqrt{48}=4\sqrt3.

Surd

A surd is an irrational root left in exact form, such as 2\sqrt2 or 7\sqrt7. A root that is rational, such as 9=3\sqrt9=3, is not a surd. The square-root symbol means the nonnegative square root.

Use square factors because their square roots are whole numbers. For example, 16 is useful in 48 because 48=16×348=16\times3.

Square root of a product

For a≥0a\ge0 and b≥0b\ge0,

ab=a b\sqrt{ab}=\sqrt a\,\sqrt b

In particular, if k≥0k\ge0, then k2b=kb\sqrt{k^2b}=k\sqrt b.

Not in the supplied formula list: learn it.

Both sides of the product rule are nonnegative and have square abab, which explains why the rule holds.

[DIAGRAM: asset_name: Square factors in a square of area forty-eight; asset_slug: surd_square_factors; caption: Four small side lengths make the large side, so the side is four times the square root of three.; description: A large square is partitioned into a four-by-four grid of sixteen equal small squares. Each small square has area three and side square root of three. The large square has area forty-eight and side four times square root of three, equal to square root of forty-eight.]
Four small side lengths make the large side, so the side is four times the square root of three.

Worked example 1: extract a square factor

Question. Simplify 48\sqrt{48}.

Plan. Write 48 as its largest square factor times the remaining factor.

48=16×316 is a square factor=163product rule=43square root of 16\begin{aligned} \sqrt{48}&=\sqrt{16\times3} &&\text{16 is a square factor}\\ &=\sqrt{16}\sqrt3 &&\text{product rule}\\ &=4\sqrt3 &&\text{square root of 16} \end{aligned}

Answer. 434\sqrt3.

Check. (43)2=16×3=48(4\sqrt3)^2=16\times3=48, and 434\sqrt3 is positive. Also, 3 has no square factor greater than 1.

The next radicand has a larger square factor. A radicand is the number inside the root sign.

Worked example 2: simplify fully

Question. Simplify 108\sqrt{108}.

Plan. Use 108=36×3108=36\times3 to remove the largest square factor in one step.

108=36×3=63\begin{aligned} \sqrt{108}&=\sqrt{36\times3}\\ &=6\sqrt3 \end{aligned}

Starting with 108=4×27108=4\times27 also works, but gives 2272\sqrt{27}, which still needs simplifying to 2×332\times3\sqrt3.

Answer. 636\sqrt3.

Check. (63)2=36×3=108(6\sqrt3)^2=36\times3=108, and 63>06\sqrt3>0.

Try the square factors 4,9,16,25,36,…4,9,16,25,36,\ldots, or use prime factors to find pairs. “Fully simplified” means no square factor greater than 1 remains inside the square root.

Watch out

Square-root multiplication does not give a matching addition rule. For example, 9+16=5\sqrt{9+16}=5, whereas 9+16=7\sqrt9+\sqrt{16}=7. Do not split a sum under a square root.

Use a square factor and check by squaring.

Like surds

Three lots of 7\sqrt7 plus five lots of 7\sqrt7 make eight lots of the same quantity. This works just as three metres plus five metres make eight metres.

Like surds

Like surds have the same root part after simplification. Their coefficients can be added or subtracted: ab+cb=(a+c)ba\sqrt b+c\sqrt b=(a+c)\sqrt b. For example, 252\sqrt5 and 757\sqrt5 are like surds; 252\sqrt5 and 232\sqrt3 are not.

Keep the root part fixed when combining coefficients. Unlike root parts cannot be combined by this rule.

Worked example 3: combine like root parts

Question. Simplify 37+57−273\sqrt7+5\sqrt7-2\sqrt7.

Plan. Combine the coefficients because every term has root part 7\sqrt7.

37+57−27=(3+5−2)7=67\begin{aligned} 3\sqrt7+5\sqrt7-2\sqrt7&=(3+5-2)\sqrt7\\ &=6\sqrt7 \end{aligned}

Answer. 676\sqrt7.

Check. Dividing both the original expression and the answer by the nonzero number 7\sqrt7 gives 3+5−2=63+5-2=6.

In the next expression, the matching root parts are hidden until you simplify each term.

Worked example 4: simplify before combining

Question. Simplify 75+12−27\sqrt{75}+\sqrt{12}-\sqrt{27}.

Plan. Extract square factors first, then combine the resulting like surds.

75+12−27=53+23−33=(5+2−3)3=43\begin{aligned} \sqrt{75}+\sqrt{12}-\sqrt{27} &=5\sqrt3+2\sqrt3-3\sqrt3\\ &=(5+2-3)\sqrt3\\ &=4\sqrt3 \end{aligned}

Answer. 434\sqrt3.

Check. Factor 3\sqrt3 from the original form: 3(25+4−9)=3(5+2−3)=43\sqrt3(\sqrt{25}+\sqrt4-\sqrt9)=\sqrt3(5+2-3)=4\sqrt3.

Some expressions contain more than one root part. For example, 23+5+43=63+52\sqrt3+\sqrt5+4\sqrt3=6\sqrt3+\sqrt5; the 5\sqrt5 term stays separate.

Watch out

75+12≠87\sqrt{75}+\sqrt{12}\ne\sqrt{87}. First simplify to 53+23=735\sqrt3+2\sqrt3=7\sqrt3. Adding radicands does not add the roots.

Simplification will reveal one common root part in this check.

Products of surds

Multiplication differs from addition: different square-root parts can be multiplied using the product rule. Multiply ordinary coefficients together and root parts together, then simplify the result.

You already know

For nonnegative a,ba,b, ab=ab\sqrt a\sqrt b=\sqrt{ab}. In particular, aa=a\sqrt a\sqrt a=a. A whole-number coefficient outside a root is not moved inside unchanged.

For example, 23×452\sqrt3\times4\sqrt5 has coefficient 2×42\times4 and root part 3×5\sqrt{3\times5}.

Worked example 5: multiply two surd terms

Question. Simplify (26)(315)(2\sqrt6)(3\sqrt{15}).

Plan. Multiply coefficients and root parts separately, then extract a square factor.

(26)(315)=690=69×10=1810\begin{aligned} (2\sqrt6)(3\sqrt{15})&=6\sqrt{90}\\ &=6\sqrt{9\times10}\\ &=18\sqrt{10} \end{aligned}

Answer. 181018\sqrt{10}.

Check. Both expressions are positive. Their squares agree: (26)2(315)2=(4×6)(9×15)=3240(2\sqrt6)^2(3\sqrt{15})^2=(4\times6)(9\times15)=3240, and (1810)2=324×10=3240(18\sqrt{10})^2=324\times10=3240.

When a bracket contains a sum or difference, multiply every term inside it. This is the distributive rule: k(p−q)=kp−kqk(p-q)=kp-kq.

Worked example 6: distribute a surd factor

Question. Simplify 5(25−3)\sqrt5(2\sqrt5-3).

Plan. Multiply 5\sqrt5 by each term, keeping the subtraction sign.

5(25−3)=255−35=2×5−35=10−35\begin{aligned} \sqrt5(2\sqrt5-3)&=2\sqrt5\sqrt5-3\sqrt5\\ &=2\times5-3\sqrt5\\ &=10-3\sqrt5 \end{aligned}

Answer. 10−3510-3\sqrt5.

Check. With 5≈2.23607\sqrt5\approx2.23607, the original expression and the answer both give approximately 3.29183.2918. The decimal is a check, not a replacement for the exact answer.

The large square at the start now fits the same rules: 43×43=16×3=484\sqrt3\times4\sqrt3=16\times3=48, so its exact side length is 434\sqrt3.

Watch out

32≠63\sqrt2\ne\sqrt6. Moving 3 inside a square root requires squaring it: 32=9×2=183\sqrt2=\sqrt{9\times2}=\sqrt{18}. Multiplication must account for the coefficient as well as the radicand.

Multiply both bracket terms and leave an exact answer.

Quick Recap

  • Extract square factors and leave no square factor greater than 1 inside the root.
  • Simplify root parts before deciding which terms are like surds.
  • Add coefficients of like surds; multiply coefficients and radicands for products.
  • Distribute a factor across every term of a bracket and retain exact root notation.

The opening square has exact side length 48=43\sqrt{48}=4\sqrt3, since (43)2=48(4\sqrt3)^2=48.