การหารเลขยกกำลังที่มีเลขชี้กำลังเป็นบวก
สมบัติการหารเลขยกกำลัง
\(\mathtt{a^m \div a^n = a^{m \; – \; n}}\)
เมื่อ a เป็นจำนวนใดๆ ที่ไม่เท่ากับ 0
m และ n เป็นจำนวนเต็มบวก
บทนิยามอื่นๆ ที่เกี่ยวข้องกับการหารเลขยกกำลัง
1. เมื่อ a เป็นจำนวนใดๆ ที่ไม่เท่ากับ 0
\(\mathtt{a^0 = 1}\)
2. เมื่อ a เป็นจำนวนใดๆ ที่ไม่เท่ากับ 0 และ n เป็นจำนวนเต็มบวก
\(\mathtt{a^{-n} = \frac{1}{a^n}}\)
Note : การหารเลขยกกำลังฐานของเลขยกกำลังต้องเท่ากันจึงจะนำเลขชี้กำลังมาลบกันได้
1. จงหาผลลัพธ์
1) \(\mathtt{2^7 \div 2^3}\)
วิธีทำ
\(\mathtt{2^7 \div 2^3}\)
= \(\mathtt{2^{7 \, – \, 3}}\)
= \(\mathtt{2^4}\)
ตอบ \(\mathtt{2^4}\)
2) \(\mathtt{(-3)^7 \div 3^4}\)
วิธีทำ
\(\mathtt{(-3)^7 \div 3^4}\)
= \(\mathtt{(-3)^7 \div (-3)^4}\)
= \(\mathtt{(-3)^{7 \, – \, 4}}\)
= \(\mathtt{(-3)^3}\)
= \(\mathtt{(-3)^3}\)
ตอบ \(\mathtt{(-3)^3}\)
3) \(\mathtt{(0.5)^4 \div (0.5)^6}\)
วิธีทำ
\(\mathtt{(0.5)^4 \div (0.5)^6}\)
= \(\mathtt{(0.5)^{4 \, – \, 6}}\)
= \(\mathtt{(0.5)^{-2}}\)
= \(\mathtt{\frac{1}{(0.5)^2}}\)
= \(\mathtt{\frac{1}{(0.5)^2}}\)
ตอบ \(\mathtt{\frac{1}{(0.5)^2}}\)
4) \(\mathtt{(-11)^5 \div (-11)^9}\)
วิธีทำ
\(\mathtt{(-11)^5 \div (-11)^9}\)
= \(\mathtt{(-11)^{5 \, – \, 9}}\)
= \(\mathtt{(-11)^{-4}}\)
= \(\mathtt{- \, \frac{1}{(11)^4}}\)
= \(\mathtt{- \, \frac{1}{(11)^4}}\)
ตอบ \(\mathtt{- \, \frac{1}{(11)^4}}\)
5) \(\mathtt{(0.8)^4 \div (\frac{4}{5})^3}\)
วิธีทำ
\(\mathtt{(0.8)^4 \div (\frac{4}{5})^3}\)
= \(\mathtt{(\frac{8}{10})^4 \div (\frac{4}{5})^3}\)
= \(\mathtt{(\frac{4}{5})^4 \div (\frac{4}{5})^3}\)
= \(\mathtt{(\frac{4}{5})^{4 \, – \, 3}}\)
= \(\mathtt{\frac{4}{5}}\)
= \(\mathtt{(\frac{4}{5})^{4 \, – \, 3}}\)
= \(\mathtt{\frac{4}{5}}\)
ตอบ \(\mathtt{\frac{4}{5}}\)
6) \(\mathtt{(\frac{1}{2})^3 \div (0.5)^4}\)
วิธีทำ
\(\mathtt{(\frac{1}{2})^3 \div (0.5)^4}\)
= \(\mathtt{(0.5)^3 \div (0.5)^4}\)
= \(\mathtt{(0.5)^{3 \, – \, 4}}\)
= \(\mathtt{(0.5)^{-1}}\)
= \(\mathtt{\frac{1}{0.5}}\)
= \(\mathtt{(0.5)^{-1}}\)
= \(\mathtt{\frac{1}{0.5}}\)
ตอบ \(\mathtt{\frac{1}{0.5}}\)
7) \(\mathtt{(0.3)^0 \div (0.3)^3}\)
วิธีทำ
\(\mathtt{(0.3)^0 \div (0.3)^3}\)
= \(\mathtt{(0.3)^{0 \, – \, 3}}\)
= \(\mathtt{(0.3)^{-3}}\)
= \(\mathtt{\frac{1}{(0.3)^3}}\)
= \(\mathtt{\frac{1}{(0.3)^3}}\)
ตอบ \(\mathtt{\frac{1}{(0.3)^3}}\)
8) \(\mathtt{(4^2 \times 4^3) \div 4^4}\)
วิธีทำ
\(\mathtt{(4^2 \times 4^3) \div 4^4}\)
= \(\mathtt{4^{2 \, + \, 3} \div 4^4}\)
= \(\mathtt{4^5 \div 4^4}\)
= \(\mathtt{4^{5 \, – \, 4}}\)
= \(\mathtt{4^1}\)
= \(\mathtt{4}\)
= \(\mathtt{4^{5 \, – \, 4}}\)
= \(\mathtt{4^1}\)
= \(\mathtt{4}\)
ตอบ 4
9) \(\mathtt{4^2 \times (4^3 \div 4^4)}\)
วิธีทำ
\(\mathtt{4^2 \times (4^3 \div 4^4)}\)
= \(\mathtt{4^2 \times 4^{3 \, – \, 4}}\)
= \(\mathtt{4^2 \times 4^{-1}}\)
= \(\mathtt{4^{2 \, + \, (-1)}}\)
= \(\mathtt{4^1}\)
= \(\mathtt{4}\)
= \(\mathtt{4^{2 \, + \, (-1)}}\)
= \(\mathtt{4^1}\)
= \(\mathtt{4}\)
ตอบ 4
10) \(\mathtt{(5^2 \times 5^3) \div 5}\)
วิธีทำ
\(\mathtt{(5^2 \times 5^3) \div 5}\)
= \(\mathtt{5^{2 \, + \, 3} \div 5}\)
= \(\mathtt{5^5 \div 5}\)
= \(\mathtt{5^{5 \, – \, 1}}\)
= \(\mathtt{5^4}\)
= \(\mathtt{5^{5 \, – \, 1}}\)
= \(\mathtt{5^4}\)
ตอบ \(\mathtt{5^4}\)
11) \(\mathtt{(m^2 \div m^3) \times m^4}\) เมื่อ \(\mathtt{m \ne 0}\)
วิธีทำ
\(\mathtt{(m^2 \div m^3) \times m^4}\)
= \(\mathtt{m^{2 \, – \, 3} \times m^4}\)
= \(\mathtt{m^{-1} \times m^4}\)
= \(\mathtt{m^{-1 \, + \, 4}}\)
= \(\mathtt{m^3}\)
= \(\mathtt{m^{-1 \, + \, 4}}\)
= \(\mathtt{m^3}\)
ตอบ \(\mathtt{m^3}\) เมื่อ \(\mathtt{m \ne 0}\)
12) \(\mathtt{(a^2 \times a^3) \div (a^0 \times a^5)}\) เมื่อ \(\mathtt{a \ne 0}\)
วิธีทำ
\(\mathtt{(a^2 \times a^3) \div (a^0 \times a^5)}\)
= \(\mathtt{a^{3 \, + \, 2} \div (1 \times a^5)}\)
= \(\mathtt{a^5 \div a^5}\)
= \(\mathtt{a^{5 \, – \, 5}}\)
= \(\mathtt{a^0}\)
= 1
= \(\mathtt{a^{5 \, – \, 5}}\)
= \(\mathtt{a^0}\)
= 1
ตอบ 1
13) \(\mathtt{(a^2 \times a) \times (a^3 \div a^5)}\) เมื่อ \(\mathtt{a \ne 0}\)
วิธีทำ
\(\mathtt{(a^2 \times a) \times (a^3 \div a^5)}\)
= \(\mathtt{a^{2 \, + \, 1} \times a^{2 \, – \, 5}}\)
= \(\mathtt{a^3 \times a^{-2}}\)
= \(\mathtt{a^{3 \, + \, (-2)}}\)
= \(\mathtt{a^1}\)
= \(\mathtt{a}\)
= \(\mathtt{a^{3 \, + \, (-2)}}\)
= \(\mathtt{a^1}\)
= \(\mathtt{a}\)
ตอบ a เมื่อ \(\mathtt{a \ne 0}\)
14) \(\mathtt{\dfrac{m^n \, \times \, m^{2n}}{m^0 \, \times \, m^{3n}}}\) เมื่อ \(\mathtt{m \ne 0}\) และ n เป็นจำนวนเต็มบวก
วิธีทำ
\(\mathtt{\dfrac{m^n \, \times \, m^{2n}}{m^0 \, \times \, m^{3n}}}\)
= \(\mathtt{\dfrac{m^{n \, + \, 2n}}{1 \, \times \, m^{3n}}}\)
= \(\mathtt{\dfrac{m^{3n}}{m^{3n}}}\)
= \(\mathtt{m^{3n \, – \, 3n}}\)
= \(\mathtt{m^0}\)
= 1
ตอบ 1