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Mathematics Class 9 Chapter 1: Real Numbers

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This is the free study page for Mathematics Class 9, Chapter 1: Real Numbers. It follows the Federal Board (FBISE) SSC-I syllabus. This chapter has 2 learning outcomes, and Parhayi.com has 241 MCQs, 63 short questions and 17 long questions for it. Read the topics and the key questions with their answers below, then tap Practice this chapter to try the questions yourself.

Mathematics grows step by step. Read the rule, study one worked example, then solve similar questions yourself and check every step.

What you will learn

Key questions and answers

These are the short questions of this chapter with their answers (63 questions).

Q1. Evaluate \(\left(\dfrac{27}{64}\right)^{-\frac23}\times4^{\frac32}\). (3 marks)

\(\left(\dfrac{27}{64}\right)^{-\frac23}=\left(\dfrac{64}{27}\right)^{\frac23}\)negative exponent: take the reciprocal
\(\left(\dfrac{4^3}{3^3}\right)^{\frac23}=\left(\dfrac43\right)^{2}=\dfrac{16}9\)power of a power: \(3\times\frac23=2\)
\(4^{\frac32}=(\sqrt4)^3=2^3=8\)rational exponent as radical
\(\dfrac{16}{9}\times8=\dfrac{128}{9}\)multiply

Q2. Find the value of \(8^{\frac23}+16^{\frac34}-25^{\frac12}\). (3 marks)

\(8^{\frac23}=(2^3)^{\frac23}=2^2=4\)power of a power
\(16^{\frac34}=(2^4)^{\frac34}=2^3=8\)power of a power
\(25^{\frac12}=5\)square root
\(4+8-5=7\)add and subtract

Q3. Simplify \(\dfrac{32^{\frac25}\times9^{\frac32}}{81^{\frac14}\times4}\). (3 marks)

\(32^{\frac25}=(2^5)^{\frac25}=2^2=4\)prime base
\(9^{\frac32}=(3^2)^{\frac32}=3^3=27\)prime base
\(81^{\frac14}=(3^4)^{\frac14}=3\)prime base
\(\dfrac{4\times27}{3\times4}=\dfrac{108}{12}=9\)cancel and divide

Q4. Evaluate \((0.04)^{-\frac32}\). (3 marks)

\(0.04=\dfrac{4}{100}=\dfrac1{25}\)decimal to fraction
\(\left(\dfrac1{25}\right)^{-\frac32}=25^{\frac32}\)negative exponent
\(25^{\frac32}=(\sqrt{25})^3=5^3=125\)root first, then power

Q5. Simplify \(\sqrt[3]{-64}\times16^{-\frac12}\). (3 marks)

\(\sqrt[3]{-64}=-4\)\((-4)^3=-64\)
\(16^{-\frac12}=\dfrac1{16^{1/2}}=\dfrac14\)negative exponent
\(-4\times\dfrac14=-1\)multiply

Q6. Find the value of \(3\left(\dfrac18\right)^{-\frac23}-\left(\dfrac14\right)^{-\frac12}\). (3 marks)

\(\left(\dfrac18\right)^{-\frac23}=8^{\frac23}=(2^3)^{\frac23}=4\)reciprocal, then power of a power
\(\left(\dfrac14\right)^{-\frac12}=4^{\frac12}=2\)reciprocal, then root
\(3\times4-2=10\)multiply and subtract

Q7. Simplify \(\left(\dfrac{x^{\frac23}y^{-\frac12}}{x^{-\frac13}y^{\frac32}}\right)^{3}\), writing the answer with positive exponents. (3 marks)

\(\dfrac{x^{\frac23}}{x^{-\frac13}}=x^{\frac23+\frac13}=x^{1}\)quotient rule
\(\dfrac{y^{-\frac12}}{y^{\frac32}}=y^{-\frac12-\frac32}=y^{-2}\)quotient rule
\(\left(x\,y^{-2}\right)^{3}=x^{3}y^{-6}\)power of a product
\(=\dfrac{x^{3}}{y^{6}}\)positive exponents

Q8. Simplify \(\dfrac{\left(a^{2}b^{-3}\right)^{\frac12}}{\left(a^{-1}b\right)^{2}}\). (3 marks)

\(\left(a^{2}b^{-3}\right)^{\frac12}=a\,b^{-\frac32}\)power of a product
\(\left(a^{-1}b\right)^{2}=a^{-2}b^{2}\)power of a product
\(\dfrac{a\,b^{-\frac32}}{a^{-2}b^{2}}=a^{1+2}\,b^{-\frac32-2}\)quotient rule
\(=a^{3}b^{-\frac72}=\dfrac{a^{3}}{b^{\frac72}}\)simplify

Q9. Write \(\sqrt[3]{x^{2}\sqrt{x}}\) in the form \(x^{r}\). (3 marks)

\(x^{2}\sqrt x=x^{2}\cdot x^{\frac12}=x^{\frac52}\)product rule
\(\sqrt[3]{x^{\frac52}}=\left(x^{\frac52}\right)^{\frac13}\)radical to exponent
\(=x^{\frac56}\)multiply exponents

Q10. Simplify \(\left(\dfrac{16a^{4}}{81b^{-8}}\right)^{\frac14}\). (3 marks)

\(\left(\dfrac{16a^{4}}{81b^{-8}}\right)^{\frac14}=\dfrac{16^{\frac14}a^{4\cdot\frac14}}{81^{\frac14}b^{-8\cdot\frac14}}\)power of a quotient and product
\(=\dfrac{2\,a}{3\,b^{-2}}\)\(16^{1/4}=2,\ 81^{1/4}=3\)
\(=\dfrac{2ab^{2}}{3}\)move \(b^{-2}\) to the numerator

Q11. Simplify \(\dfrac{\sqrt[3]{a^{5}}\cdot\sqrt{a}}{a^{\frac43}}\). (3 marks)

\(\sqrt[3]{a^5}=a^{\frac53},\quad\sqrt a=a^{\frac12}\)radical to exponent
\(\dfrac{a^{\frac53}\cdot a^{\frac12}}{a^{\frac43}}=a^{\frac53+\frac12-\frac43}\)product and quotient rules
\(\dfrac53+\dfrac12-\dfrac43=\dfrac{10+3-8}{6}=\dfrac56\)common denominator
\(=a^{\frac56}\)answer

Q12. Simplify and give the answer with positive exponents: \(\left(\dfrac{x^{-2}y^{3}}{x^{4}y^{-1}}\right)^{-\frac12}\). (3 marks)

\(\dfrac{x^{-2}y^{3}}{x^{4}y^{-1}}=x^{-6}y^{4}\)quotient rule
\(\left(x^{-6}y^{4}\right)^{-\frac12}=x^{3}y^{-2}\)power rule: \(-6\times-\frac12=3,\ 4\times-\frac12=-2\)
\(=\dfrac{x^{3}}{y^{2}}\)positive exponents

Q13. Write each in exponential form: (i) \(\sqrt[3]{x^2}\) (ii) \(\sqrt[5]{y^{-3}}\) (iii) \(\dfrac1{\sqrt[4]{z^3}}\). (3 marks)

\(\sqrt[3]{x^2}=x^{\frac23}\)\(\sqrt[n]{a^m}=a^{m/n}\)
\(\sqrt[5]{y^{-3}}=y^{-\frac35}\)same rule
\(\dfrac1{\sqrt[4]{z^3}}=\dfrac1{z^{\frac34}}=z^{-\frac34}\)reciprocal gives a negative exponent

Q14. Write each in radical form: (i) \(a^{\frac34}\) (ii) \((2b)^{\frac25}\) (iii) \(m^{-\frac13}\). (3 marks)

\(a^{\frac34}=\sqrt[4]{a^3}\)denominator = index
\((2b)^{\frac25}=\sqrt[5]{(2b)^2}=\sqrt[5]{4b^2}\)denominator = index
\(m^{-\frac13}=\dfrac1{m^{1/3}}=\dfrac1{\sqrt[3]{m}}\)negative exponent: reciprocal

Q15. Write \(\sqrt{x}\cdot\sqrt[3]{x}\cdot\sqrt[6]{x}\) as a single power of \(x\) and simplify. (3 marks)

\(\sqrt x\cdot\sqrt[3]x\cdot\sqrt[6]x=x^{\frac12}\cdot x^{\frac13}\cdot x^{\frac16}\)radicals to exponents
\(=x^{\frac12+\frac13+\frac16}\)product rule
\(\dfrac12+\dfrac13+\dfrac16=\dfrac{3+2+1}{6}=1\)common denominator
\(=x^{1}=x\)answer

Q16. Write with a smaller index: (i) \(\sqrt[6]{16}\) (ii) \(\sqrt[8]{x^{6}}\) (iii) \(\sqrt[9]{a^{3}b^{6}}\) (\(x,a,b>0\)). (3 marks)

\(\sqrt[6]{16}=\sqrt[6]{2^4}=2^{\frac46}=2^{\frac23}=\sqrt[3]{4}\)divide index and power by 2
\(\sqrt[8]{x^6}=x^{\frac68}=x^{\frac34}=\sqrt[4]{x^3}\)divide by 2
\(\sqrt[9]{a^3b^6}=\left(a^{3}b^{6}\right)^{\frac19}=a^{\frac13}b^{\frac23}=\sqrt[3]{ab^2}\)divide by 3

Q17. Show that \(\dfrac{3^{n+1}+3^{n}}{4\cdot3^{n}}=1\). (3 marks)

\(3^{n+1}+3^{n}=3^{n}\cdot3+3^{n}\)product rule
\(=3^{n}(3+1)=4\cdot3^{n}\)take \(3^n\) common
\(\dfrac{4\cdot3^{n}}{4\cdot3^{n}}=1\)cancel

Q18. Prove that \(\left(\dfrac{x^{a}}{x^{b}}\right)^{a+b}\cdot\left(\dfrac{x^{b}}{x^{c}}\right)^{b+c}\cdot\left(\dfrac{x^{c}}{x^{a}}\right)^{c+a}=1\). (3 marks)

\(\text{LHS}=\left(x^{a-b}\right)^{a+b}\cdot\left(x^{b-c}\right)^{b+c}\cdot\left(x^{c-a}\right)^{c+a}\)quotient rule
\(=x^{(a-b)(a+b)}\cdot x^{(b-c)(b+c)}\cdot x^{(c-a)(c+a)}\)power rule
\(=x^{a^2-b^2+b^2-c^2+c^2-a^2}\)product rule; \((p-q)(p+q)=p^2-q^2\)
\(=x^{0}=1=\text{RHS}\)zero exponent

Q19. Show that \(\sqrt[3]{a^{2}\sqrt a}\cdot\sqrt a\cdot a^{\frac16}=a^{\frac32}\) for \(a>0\). (3 marks)

\(a^{2}\sqrt a=a^{2+\frac12}=a^{\frac52}\)product rule
\(\sqrt[3]{a^{\frac52}}=a^{\frac52\cdot\frac13}=a^{\frac56}\)radical to exponent
\(\text{LHS}=a^{\frac56}\cdot a^{\frac12}\cdot a^{\frac16}=a^{\frac56+\frac36+\frac16}\)product rule
\(=a^{\frac96}=a^{\frac32}=\text{RHS}\)simplify

Q20. Show that \(\dfrac{9^{n}\cdot3^{2}\cdot\left(3^{-n}\right)^{2}}{3^{3}\cdot27^{-1}}=9\). (3 marks)

\(9^{n}=(3^2)^n=3^{2n}\)prime base
\(\text{Numerator}=3^{2n}\cdot3^{2}\cdot3^{-2n}=3^{2n+2-2n}=3^{2}=9\)product rule
\(\text{Denominator}=3^{3}\cdot3^{-3}=3^{0}=1\)\(27=3^3\)
\(\dfrac91=9\)divide

Q21. If \(p\neq q\), show that \(\left(\dfrac{a^{p}}{a^{q}}\right)^{\frac1{p-q}}=a\) for \(a>0\). (3 marks)

\(\dfrac{a^{p}}{a^{q}}=a^{p-q}\)quotient rule
\(\left(a^{p-q}\right)^{\frac1{p-q}}=a^{(p-q)\cdot\frac1{p-q}}\)power of a power
\(=a^{1}=a\)\(p-q\neq0\)

Q22. Find \(k\) if \(\sqrt{\dfrac{27^{2}\times3^{-1}}{9^{\frac32}}}=3^{k}\). (3 marks)

\(27^{2}=(3^3)^2=3^{6}\)prime base
\(9^{\frac32}=(3^2)^{\frac32}=3^{3}\)prime base
\(\dfrac{3^{6}\times3^{-1}}{3^{3}}=3^{6-1-3}=3^{2}\)product and quotient rules
\(\sqrt{3^{2}}=3^{1}=3^{k}\Rightarrow k=1\)equate exponents

Q23. Solve for \(x\): \(2^{x+3}=4^{x-1}\). (3 marks)

\(4^{x-1}=(2^2)^{x-1}=2^{2x-2}\)same base
\(2^{x+3}=2^{2x-2}\)rewrite
\(x+3=2x-2\)equate exponents
\(x=5\)solve

Q24. Solve for \(x\): \(5^{x}\cdot25^{x-1}=125\). (3 marks)

\(25^{x-1}=5^{2x-2}\)\(25=5^2\)
\(5^{x}\cdot5^{2x-2}=5^{3x-2}\)product rule
\(5^{3x-2}=5^{3}\)\(125=5^3\)
\(3x-2=3\Rightarrow x=\dfrac53\)equate exponents

Q25. Find \(p\) if \(\left(\dfrac{8}{27}\right)^{p}=\dfrac94\). (3 marks)

\(\dfrac8{27}=\left(\dfrac23\right)^{3}\)cube
\(\dfrac94=\left(\dfrac32\right)^{2}=\left(\dfrac23\right)^{-2}\)reciprocal
\(\left(\dfrac23\right)^{3p}=\left(\dfrac23\right)^{-2}\)same base
\(3p=-2\Rightarrow p=-\dfrac23\)equate exponents

Q26. If \(a^{x}=b\), \(b^{y}=c\) and \(c^{z}=a\) (all positive and not equal to 1), prove that \(xyz=1\). (3 marks)

\(a=c^{z}\)given
\(c=b^{y}\Rightarrow a=\left(b^{y}\right)^{z}=b^{yz}\)substitute, power rule
\(b=a^{x}\Rightarrow a=\left(a^{x}\right)^{yz}=a^{xyz}\)substitute, power rule
\(a^{1}=a^{xyz}\Rightarrow xyz=1\)equate exponents (\(a\neq1\))

Q27. Simplify \(\dfrac{4^{n+1}-4^{n}}{3\cdot4^{n}}\). (3 marks)

\(4^{n+1}-4^{n}=4^{n}\cdot4-4^{n}\)split
\(=4^{n}(4-1)=3\cdot4^{n}\)common factor
\(\dfrac{3\cdot4^{n}}{3\cdot4^{n}}=1\)cancel

Q28. Simplify \(\dfrac{2^{n}+2^{n+1}+2^{n+2}}{7\cdot2^{n-1}}\). (3 marks)

\(2^{n}+2^{n+1}+2^{n+2}=2^{n}(1+2+4)=7\cdot2^{n}\)common factor
\(\dfrac{7\cdot2^{n}}{7\cdot2^{n-1}}=2^{n-(n-1)}\)quotient rule
\(=2^{1}=2\)simplify

Q29. Simplify \(\dfrac{15^{n}\cdot6^{n}}{10^{n}\cdot9^{n}}\). (3 marks)

\(\dfrac{15^{n}\cdot6^{n}}{10^{n}\cdot9^{n}}=\left(\dfrac{15\times6}{10\times9}\right)^{n}\)power of a product/quotient
\(=\left(\dfrac{90}{90}\right)^{n}=1^{n}=1\)simplify

Q30. Simplify \(\sqrt{50}+\sqrt{18}-\sqrt{8}\). (3 marks)

\(\sqrt{50}=\sqrt{25\times2}=5\sqrt2\)perfect square factor
\(\sqrt{18}=\sqrt{9\times2}=3\sqrt2\)perfect square factor
\(\sqrt8=\sqrt{4\times2}=2\sqrt2\)perfect square factor
\(5\sqrt2+3\sqrt2-2\sqrt2=6\sqrt2\)combine like radicals

Q31. Simplify \(\sqrt[3]{24}\times\sqrt[3]{9}\). (3 marks)

\(\sqrt[3]{24}\times\sqrt[3]{9}=\sqrt[3]{24\times9}\)product rule (same index)
\(=\sqrt[3]{216}\)multiply
\(=6\)\(6^3=216\)

Q32. Simplify \((\sqrt7+\sqrt3)(\sqrt7-\sqrt3)+(\sqrt5)^2\). (3 marks)

\((\sqrt7+\sqrt3)(\sqrt7-\sqrt3)=(\sqrt7)^2-(\sqrt3)^2\)\((a+b)(a-b)=a^2-b^2\)
\(=7-3=4\)square of a root
\((\sqrt5)^2=5\)square of a root
\(4+5=9\)add

Q33. Simplify \(\dfrac{\sqrt{45}\times\sqrt{20}}{\sqrt5}\). (3 marks)

\(\dfrac{\sqrt{45}\times\sqrt{20}}{\sqrt5}=\sqrt{\dfrac{45\times20}{5}}\)product and quotient rules
\(=\sqrt{180}=\sqrt{36\times5}\)simplify
\(=6\sqrt5\)perfect square

Q34. The number of bacteria in a culture after \(t\) hours is \(P=500\times2^{\frac t4}\). Find (i) \(P\) when \(t=12\) (ii) the time when \(P=16000\). (3 marks)

\(P=500\times2^{\frac{12}{4}}=500\times2^{3}\)put \(t=12\)
\(=500\times8=4000\)simplify
\(500\times2^{\frac t4}=16000\Rightarrow2^{\frac t4}=32\)divide by 500
\(2^{\frac t4}=2^{5}\Rightarrow\dfrac t4=5\)same base
\(t=20\text{ hours}\)solve

Q35. A cube-shaped tank holds \(1728\) litres (\(1\) litre = \(1\) dm\(^3\)). Find the length of its edge and the area of its six faces. (3 marks)

\(\text{Edge}=1728^{\frac13}\)\(V=e^3\)
\(1728=12^3\Rightarrow e=12\text{ dm}\)cube root
\(\text{Area}=6e^2=6\times144\)six equal faces
\(=864\text{ dm}^2\)multiply

Q36. The length and width of a rectangular field are \(x^{\frac32}\) m and \(x^{\frac12}\) m. If \(x=16\), find its area and perimeter. (3 marks)

\(x^{\frac32}=16^{\frac32}=(\sqrt{16})^3=4^3=64\text{ m}\)length
\(x^{\frac12}=16^{\frac12}=4\text{ m}\)width
\(\text{Area}=64\times4=256\text{ m}^2\)length \(\times\) width
\(\text{Perimeter}=2(64+4)=136\text{ m}\)\(2(l+w)\)

Q37. A store has 6000 packs of 1-litre cooking oil. Every day 15% of the packs are removed. When 45% of the stock has been removed, the store must refill. (i) How many packs are removed daily? (ii) After how many days is a refill due? (3 marks)

\(15\%\text{ of }6000=\dfrac{15}{100}\times6000=900\)percentage as a fraction
\(45\%\text{ of }6000=2700\)stock to be removed before refill
\(\dfrac{2700}{900}=3\)packs removed in total \(\div\) packs removed per day
\(\text{Refill is due after }3\text{ days}\)answer

Q38. Hamza had Rs. 250,000 in his account. He wrote cheques of Rs. 78,450 and Rs. 126,375 and then deposited Rs. 40,000. Find his balance. Is the account overdrawn at any stage? (3 marks)

\(250000-78450=171550\)after the first cheque
\(171550-126375=45175\)after the second cheque
\(45175+40000=85175\)after the deposit
\(\text{Balance}=\text{Rs. }85{,}175\text{; never negative, so not overdrawn}\)answer

Q39. 4.5 metres of cloth cost Rs. 1,327.50. (i) Find the price per metre. (ii) How much will 7.2 metres cost? (3 marks)

\(1327.5\div4.5=295\)unit rate
\(7.2\times295=2124\)cost of 7.2 m
\(\text{Rs. }295\text{ per metre; Rs. }2{,}124\)answer

Q40. Ahmed drove 351.5 km using 19 litres of petrol and says his car gives about 1.85 km per litre. Is he reasonable? Find the correct average. (3 marks)

\(\text{Average}=\dfrac{351.5}{19}=18.5\text{ km/litre}\)distance \(\div\) fuel
\(\text{Estimate: }\dfrac{350}{20}=17.5\)quick check
\(1.85\neq18.5\)decimal point misplaced

Q41. A shopkeeper bought 150 kg of wheat at Rs. 85 per kg. He sold \(\frac23\) of it at Rs. 100 per kg and the rest at Rs. 80 per kg. Find his profit or loss. (3 marks)

\(\text{Cost}=150\times85=12750\)cost price
\(\tfrac23\times150=100\text{ kg sold at Rs. 100}=10000\)first sale
\(150-100=50\text{ kg sold at Rs. 80}=4000\)second sale
\(\text{Sale}=14000;\ 14000-12750=1250\)profit = selling \(-\) cost

Q42. Rs. 456,000 is to be shared by three partners in the ratio \(1:2:3\). Find each partner's share. (3 marks)

\(1+2+3=6\text{ parts}\)total parts
\(1\text{ part}=\dfrac{456000}{6}=76000\)value of one part
\(\text{Shares: }76000,\ 2\times76000=152000,\ 3\times76000=228000\)multiply
\(76000+152000+228000=456000\)check

Q43. The lowest temperature recorded at a place was \(-89.2^\circ\)C. Convert it to Fahrenheit and kelvin using \(F=\frac95C+32\) and \(K=C+273\). (3 marks)

\(F=\dfrac95(-89.2)+32\)substitute
\(=-160.56+32=-128.56\)\(\frac95\times89.2=160.56\)
\(K=-89.2+273=183.8\)substitute

Q44. Umair's monthly income is Rs. 72,000. He spends \(\frac13\) on house rent, \(\frac14\) on food and \(\frac1{12}\) on transport. How much does he save? (3 marks)

\(\dfrac13+\dfrac14+\dfrac1{12}=\dfrac{4+3+1}{12}=\dfrac8{12}=\dfrac23\)add the fractions
\(\text{Fraction saved}=1-\dfrac23=\dfrac13\)remaining part
\(\dfrac13\times72000=24000\)amount saved

Q45. A factory has assets of Rs. 3,500,000. Its debt is \(\frac27\) of the assets. Find the capital and the debt (assets = capital + debt). (3 marks)

\(\text{Debt}=\dfrac27\times3500000=1000000\)fraction of assets
\(\text{Capital}=3500000-1000000=2500000\)assets \(-\) debt

Q46. A worker is paid Rs. 85.50 per hour. He works 7.5 hours a day for 6 days. Find his weekly wage. (3 marks)

\(7.5\times6=45\text{ hours}\)total hours
\(45\times85.5=3847.5\)wage = rate \(\times\) hours

Q47. The marked price of a jacket is Rs. 4,800. A shopkeeper allows a discount of \(12\frac12\%\). If the jacket cost him Rs. 3,600, find his profit. (3 marks)

\(12\tfrac12\%=\dfrac18\)as a fraction
\(\text{Discount}=\dfrac18\times4800=600\)discount amount
\(\text{Selling price}=4800-600=4200\)after discount
\(\text{Profit}=4200-3600=600\)selling \(-\) cost

Q48. During a week a student's mobile balance changed by \(+\)Rs. 500, \(-\)Rs. 235.75, \(-\)Rs. 180.25 and \(+\)Rs. 300. If it was Rs. 62.5 at the start, find the final balance. (3 marks)

\(62.5+500=562.5\)top up
\(562.5-235.75=326.75\)first usage
\(326.75-180.25=146.5\)second usage
\(146.5+300=446.5\)second top up

Q49. Simplify \(\left[(125)^{\frac13}\times(25)^{\frac12}+(16)^{\frac34}\times(64)^{\frac13}+\dfrac{1}{(49)^{-\frac12}}\right]^{-\frac23}\) (3 marks)

\((125)^{\frac13}=5,\quad(25)^{\frac12}=5\Rightarrow5\times5=25\)prime bases
\((16)^{\frac34}=(2^4)^{\frac34}=8,\quad(64)^{\frac13}=4\Rightarrow8\times4=32\)prime bases
\(\dfrac1{(49)^{-\frac12}}=49^{\frac12}=7\)negative exponent
\(\left[25+32+7\right]^{-\frac23}=64^{-\frac23}\)add
\(64^{-\frac23}=\dfrac1{(4^3)^{\frac23}}=\dfrac1{4^2}=\dfrac1{16}\)negative exponent

Q50. Simplify: \(\dfrac{4^m\times15^{4m-2n+1}\times9^{-2m}}{10^{2m}\times25^{m-n}}\) (3 marks)

\(4^m=2^{2m};\ 9^{-2m}=3^{-4m};\ 15^{4m-2n+1}=3^{4m-2n+1}5^{4m-2n+1}\)prime factors
\(10^{2m}=2^{2m}5^{2m};\ 25^{m-n}=5^{2m-2n}\)prime factors
\(\text{Power of }2:\ 2m-2m=0\)numerator exponent \(-\) denominator exponent
\(\text{Power of }3:\ (4m-2n+1)-4m=1-2n\)same
\(\text{Power of }5:\ (4m-2n+1)-2m-(2m-2n)=1\)same
\(=3^{1-2n}\times5=\dfrac{15}{9^{n}}\)combine

Q51. Simplify the expression: \(\left[\left(2a^{\frac12}b^3\right)^4\times\left(4a^{-2}b^{-6}\right)^{\frac{-1}{2}}\right]^{\frac13}\) (3 marks)

\(\left(2a^{\frac12}b^3\right)^4=16\,a^{2}b^{12}\)power of a product
\(\left(4a^{-2}b^{-6}\right)^{-\frac12}=4^{-\frac12}a^{1}b^{3}=\dfrac12ab^3\)power of a product
\(16a^2b^{12}\times\dfrac12ab^3=8a^3b^{15}\)multiply
\(\left(8a^3b^{15}\right)^{\frac13}=2\,a\,b^{5}\)cube root

Q52. Simplify the expression: \(\left[\left(x^2y^{-3}z^4\right)^5\times\left(x^{-4}y^7z^{-2}\right)^3\right]^{\frac12}\) (3 marks)

\(\left(x^2y^{-3}z^4\right)^5=x^{10}y^{-15}z^{20}\)power rule
\(\left(x^{-4}y^{7}z^{-2}\right)^3=x^{-12}y^{21}z^{-6}\)power rule
\(\text{Product}=x^{10-12}y^{-15+21}z^{20-6}=x^{-2}y^{6}z^{14}\)product rule
\(\left[x^{-2}y^6z^{14}\right]^{\frac12}=x^{-1}y^3z^7=\dfrac{y^3z^7}{x}\)power rule

Q53. Use the law of exponents to simplify: \(\dfrac{2^{\frac13}\times27^{\frac13}\times60^{\frac12}}{180^{\frac12}\times4^{\frac{-1}{3}}\times9^{\frac14}}\) (3 marks)

\(2^{\frac13}\times27^{\frac13}\times60^{\frac12}=2^{\frac13}\cdot3\cdot60^{\frac12}\)numerator
\(180^{\frac12}\times4^{-\frac13}\times9^{\frac14}=180^{\frac12}\cdot2^{-\frac23}\cdot3^{\frac12}\)denominator
\(\dfrac{60^{\frac12}}{180^{\frac12}}=\left(\dfrac13\right)^{\frac12}=3^{-\frac12}\)combine
\(=2^{\frac13+\frac23}\cdot3\cdot3^{-\frac12}\cdot3^{-\frac12}\)collect the powers of 2 and 3
\(=2\times3\times3^{-1}=2\)simplify

Q54. Find the value of \(\sqrt{s(s-a)(s-b)(s-c)}\) if: \(a=5, b=7, c=8, s=\dfrac{a+b+c}2\) (3 marks)

\(s=\dfrac{5+7+8}{2}=10\)semi-perimeter
\(s-a=5,\ s-b=3,\ s-c=2\)differences
\(\sqrt{10\times5\times3\times2}=\sqrt{300}\)substitute
\(\sqrt{300}=\sqrt{100\times3}=10\sqrt3\)perfect square

Q55. Simplify: \(\dfrac{(2x^2)^{-4}}{(-4y^{-1}z^2)^{-2}}\) (3 marks)

\((2x^2)^{-4}=2^{-4}x^{-8}=\dfrac1{16x^{8}}\)power of a product
\(\left(-4y^{-1}z^2\right)^{-2}=(-4)^{-2}y^{2}z^{-4}=\dfrac{y^2}{16z^4}\)power of a product
\(\dfrac{1/(16x^8)}{y^2/(16z^4)}=\dfrac{16z^4}{16x^8y^2}\)divide
\(=\dfrac{z^4}{x^8y^2}\)cancel 16

Q56. Convert \(\sqrt[5]{x^3}\) to exponent form and \(x^{\frac73}\) to radical form. (3 marks)

\(\sqrt[5]{x^3}=x^{\frac35}\)index = denominator
\(x^{\frac73}=\sqrt[3]{x^7}\)denominator is the index

Q57. Using the laws of indices, find the value of \(k\) from the expression: \(\sqrt[3]{\dfrac{125^{-1}\times25^2}{\sqrt5}}=5^k\) (3 marks)

\(125^{-1}=5^{-3},\quad25^2=5^4\)prime base 5
\(\dfrac{5^{-3}\times5^{4}}{5^{\frac12}}=\dfrac{5^{1}}{5^{\frac12}}=5^{\frac12}\)product and quotient rules
\(\sqrt[3]{5^{\frac12}}=5^{\frac12\cdot\frac13}=5^{\frac16}\)cube root as exponent \(\frac13\)
\(5^{\frac16}=5^k\Rightarrow k=\dfrac16\)equate exponents

Q58. Using the laws of exponents, simplify: \(\left(\dfrac{2187\times a^5\times b^{17}}{a^{12}\times b^3}\right)^{\frac17}\) (3 marks)

\(2187=3^7\)\(3^7=2187\)
\(\dfrac{a^5}{a^{12}}=a^{-7},\quad\dfrac{b^{17}}{b^{3}}=b^{14}\)quotient rule
\(\left(3^7a^{-7}b^{14}\right)^{\frac17}\)substitute
\(=3\,a^{-1}b^{2}=\dfrac{3b^2}{a}\)power rule

Q59. Using the laws of exponents, simplify: \(\left(\dfrac{2187\times a^{5}\times b^{17}}{a^{12}\times b^{3}}\right)^{\frac{1}{7}}\) (3 marks)

\(2187=3^7\)write the number as a power of 3
\(\dfrac{a^5}{a^{12}}=a^{5-12}=a^{-7}\)law of exponents: \(\frac{a^m}{a^n}=a^{m-n}\)
\(\dfrac{b^{17}}{b^{3}}=b^{14}\)same law
\(\left(3^7\,a^{-7}\,b^{14}\right)^{\frac17}\)expression inside the bracket
\(=3^{7\times\frac17}\,a^{-7\times\frac17}\,b^{14\times\frac17}\)power of a power: \((a^m)^n=a^{mn}\)
\(=3\,a^{-1}\,b^{2}=\dfrac{3b^{2}}{a}\)final answer

Q60. Simplify \(\sqrt[3]{-125}+\sqrt{81}-16^{\frac34}\). (3 marks)

\(\sqrt[3]{-125}=-5\)\((-5)^3=-125\)
\(\sqrt{81}=9\)\(9^2=81\)
\(16^{\frac34}=(2^4)^{\frac34}=2^3=8\)law of exponents
\(-5+9-8=-4\)add the values

Q61. Rationalise the denominator of \(\dfrac{6}{\sqrt7-\sqrt5}\). (3 marks)

\(\dfrac{6}{\sqrt7-\sqrt5}\times\dfrac{\sqrt7+\sqrt5}{\sqrt7+\sqrt5}\)multiply by the conjugate
\(=\dfrac{6(\sqrt7+\sqrt5)}{7-5}\)\((a-b)(a+b)=a^2-b^2\)
\(=\dfrac{6(\sqrt7+\sqrt5)}{2}=3\sqrt7+3\sqrt5\)divide by 2

Q62. A water tank is \(\frac25\) full. After 18 litres are poured in, it is \(\frac7{10}\) full. Find the capacity of the tank. (3 marks)

Let the capacity be \(C\) litresassume the unknown
\(\frac7{10}C-\frac25C=18\)added water = difference of the two fractions
\(\frac7{10}C-\frac4{10}C=\frac3{10}C=18\)common denominator 10
\(C=18\times\frac{10}3=60\)multiply by \(\frac{10}{3}\)

Q63. Solve for \(x\): \(9^{x}\cdot3^{x-2}=27\). (3 marks)

\(9^{x}=(3^2)^x=3^{2x}\)write 9 as a power of 3
\(3^{2x}\cdot3^{x-2}=3^{3x-2}\)\(a^m\cdot a^n=a^{m+n}\)
\(27=3^3\)write 27 as a power of 3
\(3x-2=3\Rightarrow x=\frac53\)equal bases, so equal exponents

Long questions with answers

Q1. Simplify \(\left[125^{\frac23}+16^{\frac34}-36^{\frac12}\right]^{-\frac23}\). (6 marks)

\(125^{\frac23}=(5^3)^{\frac23}=5^2=25\)prime base, power of a power
\(16^{\frac34}=(2^4)^{\frac34}=2^3=8\)prime base
\(36^{\frac12}=6\)square root
\([25+8-6]^{-\frac23}=27^{-\frac23}\)add and subtract inside the bracket
\(27^{-\frac23}=\dfrac1{27^{\frac23}}=\dfrac1{(3^3)^{\frac23}}\)negative exponent
\(=\dfrac1{3^2}=\dfrac19\)power of a power

Q2. A cube-shaped tank holds 4096 litres of water (1 litre = 1 dm\(^3\)). (a) Find the length of an edge. (b) Find the area of its six faces. (c) The outside of the tank is painted at Rs. 12.50 per dm\(^2\). Find the cost. (6 marks)

\(e^{3}=4096\Rightarrow e=4096^{\frac13}\)volume of a cube
\(4096=16^3\Rightarrow e=16\text{ dm}\)(a): cube root
\(\text{Area}=6e^{2}=6\times16^{2}=6\times256\)(b): six equal faces
\(=1536\text{ dm}^2\)multiply
\(\text{Cost}=1536\times12.5\)(c): area \(\times\) rate
\(=19200\text{ rupees}\)multiply

Q3. The number of bacteria in a culture after \(t\) hours is \(N=400\times2^{\frac t2}\). (a) Find the number after 6 hours. (b) After how many hours will there be 12,800 bacteria? (c) By what factor does the number grow every 2 hours? (6 marks)

\(N=400\times2^{\frac62}=400\times2^{3}=400\times8\)(a): \(t=6\)
\(=3200\)simplify
\(400\times2^{\frac t2}=12800\Rightarrow2^{\frac t2}=32\)(b): divide by 400
\(2^{\frac t2}=2^{5}\Rightarrow\dfrac t2=5\Rightarrow t=10\text{ hours}\)same base
\(N(t+2)=400\times2^{\frac{t+2}{2}}=400\times2^{\frac t2}\times2^{1}\)(c): \(t\to t+2\)
\(\dfrac{N(t+2)}{N(t)}=2\)the number doubles

Q4. The table shows the transactions in Ayesha's bank account in October. (a) Find the balance after each transaction. (b) On which date does the account first become overdrawn? (c) How much must she deposit to bring the balance back to Rs. 25,000 on 20 October? (6 marks)

\(180000-45790=134210\)(a): after 4 Oct
\(134210-112921=21289\)(a): after 9 Oct
\(21289+60500=81789\)(a): after 15 Oct
\(81789-98000=-16211\)(a): after 20 Oct
\(\text{The balance is negative first on 20 Oct}\)(b): all earlier balances are positive
\(25000-(-16211)=41211\)(c): deposit needed

Practice MCQs

Choose the answer in your mind first. The answer is shown below each question.

1. In the radical \(\sqrt[5]{x^3y}\), the number \(5\) is called the:

  1. index of the radical
  2. base
  3. exponent
  4. radicand

Answer: a

In \(\sqrt[n]{a}\) the small number \(n\) is the index; the expression under the sign is the radicand.

2. Laiba checked four statements about indices and radicals. Which one is NOT true?

  1. \(3^{-2}=-9\)
  2. \(2^{\frac12}\cdot2^{\frac12}=2\)
  3. \(\sqrt{50}=5\sqrt2\)
  4. \(16^{-\frac14}=\dfrac12\)

Answer: a

\(\sqrt{3}\cdot\sqrt[3]{3}=3^{\frac12+\frac13}=3^{5/6}\); the exponents are added, not multiplied.

3. Find the value of \(\left(\frac{9}{25}\right)^{-\frac{3}{2}}\).

  1. \(\frac{9}{5}\)
  2. \(\frac{125}{27}\)
  3. \(\frac{3}{5}\)
  4. \(\frac{27}{125}\)

Answer: b

Negative exponent: first take the reciprocal, then the root: \(\left(\frac{9}{25}\right)^{-\frac{3}{2}}=\left(\frac{25}{9}\right)^{\frac{3}{2}}=\frac{125}{27}\).

4. Simplify \(\sqrt[3]{54}\).

  1. \(6\)
  2. \(3 \sqrt[3]{2}\)
  3. \(6 \sqrt[3]{3}\)
  4. \(9 \sqrt[3]{2}\)

Answer: b

\(54=27\times2\), so \(\sqrt[3]{54}=3\sqrt[3]{2}\).

5. A bank account shows a balance of \(-\)Rs. \(2{,}000\). What does this tell us?

  1. The account is closed
  2. The account holder owes Rs. 2,000 to the bank
  3. The bank owes Rs. 2,000 to the account holder
  4. The account holder saved Rs. 2,000

Answer: b

A negative balance means the account is overdrawn: money is owed.

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