Base and exponent pdf with answers 7th t1s11

In mathematics, exponents are a way to explicit repeated multiplication of various by way of itself. The wide variety being extended is called the base, and the wide variety of times it’s far elevated is known as the exponent. Understanding how to paintings with bases and exponents is essential for solving extra complex mathematical troubles.

Basics of Bases and Exponents

• Base (b): The quantity that is being multiplied.
• Exponent (n): The variety of instances the bottom is multiplied via itself.

The wellknown shape of a base and exponent is written as bnb^nbn, in which:

• bbb is the bottom.
• nnn is the exponent.

For example, 232^323 method 2×2×22 instances 2 instances 22×2×2, which equals 8.

Examples

1. 323^232

• Base: 3
• Exponent: 2
• Calculation: 3×3=93 instances 3 = ninety three×three=nine

1. 535^353

• Base: 5
• Exponent: 3
• Calculation: five×five×5=1255 instances five times five = 1255×five×five=125

Properties of Exponents

1. Multiplying Powers with the Same Base

• Rule: bm×bn=bm+nb^m times b^n = b^m+nbm×bn=bm+n
• Example: 23×22=23+2=25=322^three times 2^2 = 2^three+2 = 2^5 = 3223×22=23+2=25=32

1. Dividing Powers with the Same Base

• Rule: bmbn=bm−nfracb^mb^n = b^m-nbnbm​=bm−n
• Example: 3432=34−2=32=9frac3^43^2 = 3^4-2 = three^2 = 93234​=34−2=32=9

1. Power of a Power

• Rule: (bm)n=bm×n(b^m)^n = b^m instances n(bm)n=bm×n
• Example: (23)2=23×2=26=64(2^3)^2 = 2^three instances 2 = 2^6 = sixty four(23)2=23×2=26=sixty four

1. Zero Exponent

• Rule: b0=1b^zero = 1b0=1 (where b≠0b neq 0b=0)
• Example: 50=15^zero = 150=1

1. Negative Exponent

• Rule: b−n=1bnb^-n = frac1b^nb−n=bn1​
• Example: 2−3=123=182^-three = frac12^3 = frac182−3=231​=81​

Practice Problems and Solutions

1. Problem: 42×434^2 instances 4^342×43

• Solution: 42+three=45=10244^2+three = 4^5 = 102442+3=forty five=1024

1. Problem: 6562frac6^56^26265​

• Solution: sixty five−2=63=2166^5-2 = 6^three = 21665−2=63=216

1. Problem: (32)three(3^2)^three(32)3

• Solution: 32×three=36=7293^2 times 3 = 3^6 = 72932×three=36=729

1. Problem: 707^070

• Solution: 70=17^0 = one hundred seventy=1

1. Problem: five−25^-2five−2

• Solution: 5−2=152=1255^-2 = frac15^2 = frac1255−2=521​=251​

Real-World Applications

Exponents are used in numerous actual-world packages, consisting of:

• Scientific Notation: Used to explicit very big or very small numbers. For instance, the speed of mild is about 3×1083 instances 10^eighty three×108 meters per 2nd.
• Compound Interest: In finance, compound hobby calculations involve exponents. For example, the formula for compound hobby is A=P(1+rn)ntA = P(1 + fracrn)^ntA=P(1+nr​)nt, wherein PPP is the fundamental amount, rrr is the once a year interest charge, nnn is the range of times hobby is compounded consistent with yr, and ttt is the number of years.

Conclusion

Understanding bases and exponents is essential for studying more superior mathematical principles. By working towards the regulations and residences of exponents, students can expand a sturdy basis for their future research in mathematics and technology.