Irrational numbers

(\sqrt{98}=7\sqrt{2}) and (\sqrt{18}=3\sqrt{2}) so division gives (4). First convert the numerator into like radicals.

Option B is non-terminating and non-repeating, so it represents an irrational number. In contrast, 0.636363... repeats a fixed block and is rational. Exam tip: identify irrational decimals by checking for no repeating pattern.

A rational number has a terminating or repeating decimal expansion. Here, the number of zeros between successive 1s increases as 1, 2, 3, 4…, so no block repeats. Exam tip: check repetition, not merely the digits used.

The governing concept is reducing radicals to like surds before adding or subtracting them. Since 8 = 4×2, √8 = 2√2; since 18 = 9×2, √18 = 3√2. Thus m = 2√2 + 3√2 = 5√2. Also, 50 = 25×2, so n = √50 = 5√2. Consequently, m − n = 5√2 − 5√2 = 0, and option A is correct. Option B may result from subtracting an unrelated leftover radical. Option C reflects incomplete simplification or an arithmetic error. Option D is the common value of m and n, not their difference. The cancellation is exact because equal surds with equal coefficients have opposite signs in the subtraction.

Answer: 2√2 - 1, so option B is correct. To simplify x=1/(√2+1), rationalise the denominator by multiplying the numerator and denominator by the conjugate of √2+1. Its conjugate is √2-1. Thus x=[1(√2-1)]/[(√2+1)(√2-1)]. The denominator is a difference of squares: (√2)^2-1^2=2-1=1. Therefore x=√2-1. Now add √2: x+√2=(√2-1)+√2=2√2-1. Option A, 1, incorrectly treats the radical terms as if they cancel. Option C, 2, loses both the radical and the constant term. Option D is just the original denominator and is not the simplified value of the expression. The conjugate method works because (a+b)(a-b)=a²-b², which removes the surd from the denominator. Always change the sign between the two terms when choosing a conjugate.

The governing concept is extraction of perfect-square factors from radicals followed by combining like surds. Since 80 = 16×5, √80 = 4√5. Since 45 = 9×5, √45 = 3√5. Thus u = 4√5 − 3√5 = √5. The question gives v = √5, so u + v = √5 + √5 = 2√5. Therefore option A is correct. Option B results from adding coefficients without first evaluating u, option C is only the value of u and not u + v, and option D would require opposite signs. The important rule is that coefficients of identical surds may be added, while unlike radicals cannot be combined directly.

The governing concept is the difference-of-squares identity, (a+b)(a−b)=a²−b². Let a = √8 and b = √18. Then the product is (√8)² − (√18)² = 8 − 18 = −10. Therefore option A is correct. A second check is possible by simplifying first: √8 = 2√2 and √18 = 3√2. The product becomes (5√2)(−√2) = −5×2 = −10. Option B reverses the subtraction, option C adds 8 and 18 instead of subtracting them, and option D incorrectly multiplies radicands while ignoring the conjugate structure. Recognising the identity makes the calculation shorter and safer.

Answer: option A, (7 + 2√10)/3. Start with x/y = (√5 + √2)/(√5 − √2). The denominator contains a difference of square roots, so multiply the numerator and denominator by its conjugate, √5 + √2. The denominator becomes (√5 − √2)(√5 + √2) = (√5)² − (√2)² = 5 − 2 = 3. The numerator becomes (√5 + √2)² = 5 + 2 + 2√10 = 7 + 2√10. Therefore x/y = (7 + 2√10)/3. Option B is missing the denominator 3. Option C has the wrong sign for the middle term; squaring a sum gives a positive cross term. Option D is only the rationalised denominator, not the complete quotient. Since √5 − √2 is positive and nonzero, the division is valid. Memory cue: use the conjugate, and remember (a+b)² has +2ab.

The governing concept is simplifying radicals by taking perfect-square factors outside the radical and then subtracting like surds. Since 125 = 25×5, √125 = 5√5. Since 45 = 9×5, √45 = 3√5. Thus u = 5√5 − 3√5 = 2√5. Given v = √5, we obtain u − v = 2√5 − √5 = √5. Hence option A is correct. Option B incorrectly treats u and v as equal, option C stops before subtracting v, and option D adds coefficients rather than performing the required subtraction. Once the common radical is identified, only its coefficients need to be operated on.

The governing idea is rationalisation using the conjugate of a binomial surd. Since t = √7 + 2, we have 3/t = 3/(√7 + 2). Multiply numerator and denominator by √7 − 2: 3(√7 − 2)/[(√7 + 2)(√7 − 2)] = 3(√7 − 2)/(7 − 4) = √7 − 2. Therefore t + 3/t = (√7 + 2) + (√7 − 2) = 2√7. Option A is correct. Option B incorrectly removes the radical, option C fails to cancel the constants, and option D omits one of the equal √7 terms. The conjugate works because it converts the denominator into the rational number 3.

The governing concept is the difference-of-squares identity, (a+b)(a−b)=a²−b². Set a=√20 and b=√45. Then the product is (√20)²−(√45)² = 20−45 = −25, so option A is correct. The result can be checked by simplifying first: √20=2√5 and √45=3√5. The two factors become 5√5 and −√5, whose product is −5×5 = −25. Option B reverses the subtraction order. Option C adds the radicands instead of applying the identity. Option D incorrectly combines the radicands and does not represent the given conjugate product. Recognising the conjugate pattern avoids unnecessary expansion and also shows why the middle terms cancel.

The governing concept is rationalisation of a surd in the denominator. The conjugate of √11+3 is √11−3. Multiply numerator and denominator by this conjugate: r = (√11−3)/[(√11+3)(√11−3)] = (√11−3)/[(√11)²−3²] = (√11−3)/(11−9) = (√11−3)/2. Thus option A is correct. Option B has the correct numerator but loses the denominator 2. Option C retains the wrong sign and therefore uses the original expression rather than its conjugate. Option D is the negative of the numerator and is not equivalent to the original positive fraction. The denominator is nonzero, so multiplying by the conjugate preserves equality and produces a rational denominator.

The governing concept is the conjugate-product identity (u+v)(u−v)=u²−v². Since multiplication is commutative, rewrite ab as (√5+√3)(√5−√3). Taking u=√5 and v=√3 gives ab=(√5)²−(√3)²=5−3=2. Therefore option A is correct. The cross terms cancel because one product is +√15 and the other is −√15. Option B has the wrong sign and would correspond to reversing the order of the squares. Option C comes from an incorrect expansion or addition. Option D multiplies the radicals but fails to include the cancellation of the two cross terms. The conjugate structure is the important observation, and it gives an exact result without decimal approximation.

The governing concept is the conjugate-product identity (a+b)(a−b)=a²−b². Let a=√45 and b=√5. Then (√45+√5)(√45−√5)=(√45)²−(√5)²=45−5=40, so option A is correct. An independent check is √45=3√5; the factors then become 4√5 and 2√5, whose product is 8×5=40. Option B results from adding 45 and 5 rather than finding their difference. Option C reverses or mishandles the subtraction. Option D is not equivalent: √225=15, so 2√225=30, not 40. The conjugate pattern is preferable to full expansion because the positive and negative cross terms cancel immediately.

The governing concept is the square-root identity √x × √x = x for every non-negative real number x. Here, the quantity inside both identical square-root signs is 6 + √5. Since √5 is approximately 2.236, the radicand is positive, so the identity applies directly: √(6 + √5) × √(6 + √5) = (√(6 + √5))² = 6 + √5. Therefore, option A is correct. Option B incorrectly changes the plus sign to a minus sign, option C treats the expression as 6² + (√5)², and option D has no valid connection with the given product.

The governing concept is the difference-of-squares identity for conjugate expressions: (a+b)(a−b)=a²−b². Let a=√45 and b=√20. The required value is therefore (√45)²−(√20)²=45−20=25, making option A correct. Principal square roots are non-negative, and squaring each one returns its radicand exactly. Option B would result from reversing the subtraction order, but the given factors are arranged as a+b followed by a−b. Option C incorrectly adds the radicands. Option D has no valid connection with the identity and comes from an unrelated manipulation. One may also simplify the radicals to 3√5 and 2√5; the factors become 5√5 and √5, whose product is again 25.

The governing concept is rationalisation of a denominator containing a surd. The conjugate of √17 + 4 is √17 − 4. Multiply numerator and denominator by that conjugate: r = [1/(√17 + 4)]×[(√17 − 4)/(√17 − 4)]. The denominator becomes (√17 + 4)(√17 − 4) = (√17)² − 4² = 17 − 16 = 1. Hence r = (√17 − 4)/1 = √17 − 4, so option A is correct. Option B is the original denominator expression, option C reverses the sign and is negative, and option D leaves a surd in the denominator rather than rationalising it. Since √17 is slightly greater than 4, the final value is positive, confirming the sign.

The governing concept is simplifying surds by extracting perfect-square factors and then combining like surds. Since 8=4×2, √8=2√2. Since 18=9×2, √18=3√2. Therefore y=2√2+3√2=5√2. Dividing by √2 gives y/√2=(5√2)/√2=5, because √2 is non-zero. Hence option C is correct. Option A or B may result from simplifying only one radical or losing a coefficient, while option D can arise from adding coefficients incorrectly. The radicands 8 and 18 must not be added directly; both radicals first need to be expressed as multiples of the same irrational factor √2.

The governing concept is expressing all radicals with a common square-free factor before adding and dividing. We have √18=√(9×2)=3√2, √50=√(25×2)=5√2, and √8=√(4×2)=2√2. Thus the numerator is 3√2+5√2=8√2. The entire expression becomes (8√2)/(2√2)=8/2=4, because √2 is a common nonzero factor. Therefore option B is correct. Option A may result from using only part of the numerator. Option C can arise from adding radicands or coefficients incorrectly. Option D ignores the denominator. The essential rule is that radicals should first be reduced to like surds; only then can their coefficients be combined and the common factor cancelled safely.

The governing concepts are the area formula for a rectangle and the difference-of-squares identity. The area is length multiplied by breadth, so A = (√18 + √2)(√18 − √2). Let a = √18 and b = √2. Using (a + b)(a − b) = a² − b², we obtain A = (√18)² − (√2)² = 18 − 2 = 16. Therefore, option B is correct. The irrational-looking dimensions form a conjugate pair, so the radical terms cancel in the product and the area becomes the rational number 16. Option A incorrectly adds 18 and 2. Options C and D retain radicals even though the identity removes them. Both dimensions are positive, so the result is a valid positive area.

The governing concept is reducing all radicals to like surds before combining coefficients. Since 98=49×2, √98=7√2. Since 50=25×2, √50=5√2. Substitution gives u=3√2+7√2−5√2=(3+7−5)√2=5√2. Therefore u/√2=(5√2)/√2=5, as √2 is non-zero. Option C is correct. The subtraction sign must remain attached to the coefficient of √50. Options A, B and D may result from simplifying √98 or √50 incorrectly, ignoring the minus sign, or combining the original radicands rather than their coefficients. Writing every term as a multiple of √2 makes the calculation clear and reliable.

The governing concept is simplification of surds by extracting perfect-square factors. Since 108=36×3, √108=6√3. Since 300=100×3, √300=10√3. Thus s=6√3+10√3=16√3. Dividing by √3 gives s/√3=(16√3)/√3=16, because √3 is non-zero. Hence option C is correct. The irrational factor cancels only after both radicals have been expressed as like surds. Option A could result from losing one coefficient, while options B and D do not agree with the coefficient sum 6+10. Directly adding 108 and 300 would also be invalid, because square roots cannot generally be combined by adding their radicands.

The governing concept is simplification of surds by extracting perfect-square factors. Since 27 = 9 × 3, √27 = √9 × √3 = 3√3. Similarly, 75 = 25 × 3, so √75 = 5√3. Therefore y = 3√3 + 5√3 = 8√3, because like surds are added by adding their rational coefficients. Dividing by √3 gives y/√3 = 8√3/√3 = 8, since √3 is non-zero. Thus option B is correct. Option A or C may result from adding the radicands or simplifying one radical incorrectly, while option D has no valid basis. The essential method is to express both radicals with the same √3 factor before combining them.

The governing concept is rationalisation of a denominator containing a binomial surd. The conjugate of √17 + √8 is √17 − √8, so multiply both numerator and denominator by that conjugate. This gives 3(√17 − √8)/[(√17 + √8)(√17 − √8)]. Using the difference-of-squares identity, the denominator becomes (√17)² − (√8)² = 17 − 8 = 9. Hence the expression is 3(√17 − √8)/9 = (√17 − √8)/3. Therefore option C is correct. Option A misses the factor 1/3, option B leaves the denominator irrational and does not rationalise it, and option D is not algebraically equivalent. The conjugate is essential because it changes the denominator into a rational number.

The governing concept is extraction of perfect-square factors from radicals followed by addition of like surds. Since 242 = 121 × 2, √242 = 11√2. Also, 128 = 64 × 2, so √128 = 8√2. Therefore s = 11√2 + 8√2 = 19√2. Dividing by √2 gives s/√2 = 19√2/√2 = 19, because √2 is non-zero. Hence option B is correct. The values 17, 21, and 23 can arise from errors such as subtracting or misadding the coefficients, or extracting an incorrect square factor. It is not valid to add 242 and 128 inside one square root; each radical must first be simplified separately.