Exponent Calculator Tool Free – Class 8-10 Maths

Exponent Calculator Tool

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Exponent Calculator Tool Free – Class 8-10 Maths

Detailed Explanation of Exponents in Mathematics (Class 8 Level)

Exponents (also called powers) are a short and convenient way to write repeated multiplication of the same number. Instead of writing $2 \times 2 \times 2 \times 2 \times 2$2×2×2×2×2, we write $2^5$25.

• Base = the number that is being multiplied repeatedly (here 2).
• Exponent (or power) = the number of times the base is multiplied by itself (here 5). So, $a^n$an is read as “$a$a raised to the power $n$n” or “$a$a to the $n$nth power” and means $a \times a \times a \times \dots$a×a×a×… (n times).Exponent Calculator Tool

Important Cases

• Any non-zero number raised to power 0 is 1: $a^0 = 1$a0=1 (where $a \neq 0$a=0).
• Negative exponents give reciprocals: $a^{-m} = \frac{1}{a^m}$a−m=am1​ (where $a \neq 0$a=0).

Laws of Exponents (these hold for all integers m and n, and for a, b ≠ 0)

1. Product Law (Multiplication): $a^m \times a^n = a^{m+n}$am×an=am+n
2. Quotient Law (Division): $a^m \div a^n = a^{m-n}$am÷an=am−n (when $m > n$m>n) or $\frac{a^m}{a^n} = a^{m-n}$anam​=am−n
3. Power of a Power Law: $(a^m)^n = a^{m \times n}$(am)n=am×n
4. Product of Powers with Same Exponent: $a^m \times b^m = (a \times b)^m$am×bm=(a×b)m
5. Quotient of Powers with Same Exponent: $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$bmam​=(ba​)m
6. Zero Exponent Law: $a^0 = 1$a0=1 (a ≠ 0)
7. Negative Exponent Law: $a^{-m} = \frac{1}{a^m}$a−m=am1​

These laws allow us to simplify complicated expressions quickly without writing out long multiplications.

Standard Form (Scientific Notation) Very large or very small numbers are written as $k \times 10^n$k×10n, where $1 \leq k < 10$1≤k<10 and $n$n is an integer.

• For large numbers → positive exponent.
• For small numbers → negative exponent.Exponent Calculator Tool.Exponent Calculator Tool

Examples

• 3 45 00 000 = $3.45 \times 10^8$3.45×108
• 0.000 067 = $6.7 \times 10^{-5}$6.7×10−5

Now, here are 50 numerical problems (exactly from the style and difficulty of Class 8 Exponents and Powers chapter – NCERT/CBSE). Each problem is followed by its step-by-step solution using the laws above.Exponent Calculator Tool.Exponent Calculator Tool

1–10: Simplification using Product Law

1. Simplify: $2^4 \times 2^3$24×23 Solution: $2^{4+3} = 2^7 = 128$24+3=27=128
2. Simplify: $3^5 \times 3^2$35×32 Solution: $3^{5+2} = 3^7 = 2187$35+2=37=2187
3. Simplify: $5^2 \times 5^6$52×56 Solution: $5^{2+6} = 5^8 = 390625$52+6=58=390625
4. Simplify: $7^3 \times 7^1$73×71 Solution: $7^{3+1} = 7^4 = 2401$73+1=74=2401
5. Simplify: $4^4 \times 4^4$44×44 Solution: $4^{4+4} = 4^8 = 65536$44+4=48=65536
6. Simplify: $10^3 \times 10^5$103×105 Solution: $10^{3+5} = 10^8 = 100000000$103+5=108=100000000
7. Simplify: $9^2 \times 9^3$92×93 Solution: $9^{2+3} = 9^5 = 59049$92+3=95=59049
8. Simplify: $6^1 \times 6^7$61×67 Solution: $6^{1+7} = 6^8 = 1679616$61+7=68=1679616
9. Simplify: $8^4 \times 8^2$84×82 Solution: $8^{4+2} = 8^6 = 262144$84+2=86=262144
10. Simplify: $11^3 \times 11^2$113×112 Solution: $11^{3+2} = 11^5 = 161051$113+2=115=161051.Exponent Calculator Tool

11–20: Simplification using Quotient Law

11. Simplify: $2^7 \div 2^3$27÷23 Solution: $2^{7-3} = 2^4 = 16$27−3=24=16
12. Simplify: $3^6 \div 3^2$36÷32 Solution: $3^{6-2} = 3^4 = 81$36−2=34=81
13. Simplify: $5^5 \div 5^1$55÷51 Solution: $5^{5-1} = 5^4 = 625$55−1=54=625
14. Simplify: $7^4 \div 7^3$74÷73 Solution: $7^{4-3} = 7^1 = 7$74−3=71=7
15. Simplify: $4^8 \div 4^5$48÷45 Solution: $4^{8-5} = 4^3 = 64$48−5=43=64
16. Simplify: $10^9 \div 10^4$109÷104 Solution: $10^{9-4} = 10^5 = 100000$109−4=105=100000
17. Simplify: $9^5 \div 9^3$95÷93 Solution: $9^{5-3} = 9^2 = 81$95−3=92=81
18. Simplify: $6^6 \div 6^2$66÷62 Solution: $6^{6-2} = 6^4 = 1296$66−2=64=1296.Exponent Calculator Tool
19. Simplify: $8^7 \div 8^4$87÷84 Solution: $8^{7-4} = 8^3 = 512$87−4=83=512
20. Simplify: $11^4 \div 11^1$114÷111 Solution: $11^{4-1} = 11^3 = 1331$114−1=113=1331

21–30: Power of a Power Law and Mixed Rules

21. Simplify: $(2^3)^2$(23)2 Solution: $2^{3 \times 2} = 2^6 = 64$23×2=26=64
22. Simplify: $(3^4)^3$(34)3 Solution: $3^{4 \times 3} = 3^{12} = 531441$34×3=312=531441
23. Simplify: $(5^2)^4$(52)4 Solution: $5^{2 \times 4} = 5^8 = 390625$52×4=58=390625
24. Simplify: $(7^1)^5$(71)5 Solution: $7^{1 \times 5} = 7^5 = 16807$71×5=75=16807
25. Simplify: $(4^3)^2$(43)2 Solution: $4^{3 \times 2} = 4^6 = 4096$43×2=46=4096
26. Simplify: $2^3 \times 2^5 \div 2^2$23×25÷22 Solution: $2^{3+5-2} = 2^6 = 64$23+5−2=26=64.Exponent Calculator Tool
27. Simplify: $(3^2 \times 3^4)^2$(32×34)2 Solution: First $3^{2+4} = 3^6$32+4=36, then $(3^6)^2 = 3^{12} = 531441$(36)2=312=531441
28. Simplify: $(5^3)^2 \div 5^4$(53)2÷54 Solution: $5^{3 \times 2} \div 5^4 = 5^6 \div 5^4 = 5^{6-4} = 5^2 = 25$53×2÷54=56÷54=56−4=52=25
29. Simplify: $(7^2)^3 \times 7^1$(72)3×71 Solution: $7^{2 \times 3} \times 7^1 = 7^6 \times 7^1 = 7^7 = 823543$72×3×71=76×71=77=823543
30. Simplify: $(2^4 \times 3^2)^2$(24×32)2 Solution: $2^{4 \times 2} \times 3^{2 \times 2} = 2^8 \times 3^4 = 256 \times 81 = 20736$24×2×32×2=28×34=256×81=20736

31–35: Zero and Negative Exponents

31. Evaluate: $5^0$50 Solution: 1
32. Evaluate: $10^0$100 Solution: 1
33. Evaluate: $2^{-3}$2−3 Solution: $\frac{1}{2^3} = \frac{1}{8}$231​=81​
34. Simplify: $3^{-2} \times 3^4$3−2×34 Solution: $3^{-2+4} = 3^2 = 9$3−2+4=32=9
35. Simplify: $7^{-1} \div 7^{-3}$7−1÷7−3 Solution: $7^{-1 – (-3)} = 7^{2} = 49$7−1−(−3)=72=49

36–40: Rules with Different Bases

36. Simplify: $2^3 \times 3^3$23×33 Solution: $(2 \times 3)^3 = 6^3 = 216$(2×3)3=63=216
37. Simplify: $\frac{5^4}{5^2}$5254​ (already same base, but mixed) Solution: $5^{2} = 25$52=25 (or use quotient law).Exponent Calculator Tool
38. Simplify: $(4^2 \times 5^2)$(42×52) Solution: $(4 \times 5)^2 = 20^2 = 400$(4×5)2=202=400
39. Simplify: $\frac{6^3}{3^3}$3363​ Solution: $\left(\frac{6}{3}\right)^3 = 2^3 = 8$(36​)3=23=8
40. Simplify: $2^4 \times 3^4 \div 6^4$24×34÷64 Solution: $\frac{2^4 \times 3^4}{6^4} = \frac{(2 \times 3)^4}{6^4} = \frac{6^4}{6^4} = 1$6424×34​=64(2×3)4​=6464​=1

41–50: Standard Form (Scientific Notation)

41. Express 4500000 in standard form. Solution: $4.5 \times 10^6$4.5×106
42. Express 123000000 in standard form. Solution: $1.23 \times 10^8$1.23×108
43. Express 0.000078 in standard form. Solution: $7.8 \times 10^{-5}$7.8×10−5
44. Express 0.00000056 in standard form. Solution: $5.6 \times 10^{-7}$5.6×10−7.Exponent Calculator Tool
45. Convert $3.2 \times 10^5$3.2×105 to ordinary number. Solution: 320000
46. Convert $8.9 \times 10^{-4}$8.9×10−4 to ordinary number. Solution: 0.00089
47. Simplify and write in standard form: $(2 \times 10^3) \times (3 \times 10^4)$(2×103)×(3×104) Solution: $6 \times 10^7$6×107
48. Simplify and write in standard form: $(4 \times 10^6) \div (2 \times 10^3)$(4×106)÷(2×103) Solution: $2 \times 10^3$2×103
49. Express 567890000 in standard form. Solution: $5.6789 \times 10^8$5.6789×108
50. Express 0.0000001234 in standard form. Solution: $1.234 \times 10^{-7}$1.234×10−7.Exponent Calculator Tool

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