X = 4 × 10m Y = 2 × 10n
Where m and n are integers.
XY= 8 × 102 and x/y = 2 × 108
Find the value of m and the value of n
$\displaystyle XY = 8 \times 102$
$\displaystyle (4 \times 10m)(2 \times 10n) = 8 \times 102$
$\displaystyle 8 \times 100 \times mn = 8 \times 102$
$\displaystyle mn = \frac{102}{100}$ .........eqn(1)
$\displaystyle \frac{X}{Y} = 2 \times 108$
$\displaystyle \frac{4 \times 10m}{2 \times 10n} = 2 \times 108$
$\displaystyle \frac{2m}{n} = 2 \times 108$
$\displaystyle \frac{m}{n} =108$
$\displaystyle m = 108 n$ ..............eqn(2)
Put this value of m in eqn (1),
$\displaystyle (108n)n = \frac{102}{100}$
$\displaystyle 108n^2 = \frac{102}{100}$
$\displaystyle n^2 = \frac{102}{100 \times 108}$
$\displaystyle n= \sqrt{\frac{102}{100 \times 108}} $, now, simplify to get integer value
put this value of n in eqn(1),
$\displaystyle m = 108 \times \sqrt{\frac{102}{100 \times 108}}$
$\displaystyle m = \sqrt{\frac{108 \times 102}{100}}$, now, simplify to get integer value
x=4X10^m and y=2X10^n
xy=8X10^2
hence (4X10^m)(2X10^n) = 8X10^2
so 8X10^(m+n) = 8X10^2
simplified: 10^(m+n) = 10^2 (divide by 8)
hence m+n = 2
x/y=2X10^8
hence (4X10^m)/(2X10^n) = 2X10^8
so 2X10^(m-n) = 2X10^8
simplified: 10^(m-n) = 10^8 (divide by 2)
hence m-n = 8
now m+n = 2
+ m-n = 8
_____________
2m = 10 (since n-n = 0)
hence m = 5 (since 2X5 = 10)
substitute into m+n = 2
5+n = 2
n = 2 - 5
n = -3
so: m=5 and n=-3
Lovely question. Hope this helps. Cheers.
$\displaystyle XY = 8 \times 10^2$
$\displaystyle (4 \times 10^m)(2 \times 10^n) = 8 \times 10^2$
$\displaystyle 8 \times 10^{m+n} = 8 \times 10^2$
$\displaystyle 10^{m+n} = 10^2$
$\displaystyle m+n = 2 $ .........eqn(1)
Now,
$\displaystyle \frac{X}{Y} = 2 \times 10^8$
$\displaystyle \frac{4 \times 10^m}{2 \times 10^n} = 2 \times 10^8$
$\displaystyle 2\times 10^{m-n}= 2 \times 10^8$
$\displaystyle 10^{m-n}=10^8$
$\displaystyle m-n=8$ ................eqn(2)
Adding equations (1) and (2)
$\displaystyle m+n = 2 $
$\displaystyle m-n=8$
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2m = 10
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m = 5
Put this value of m in eqn(1),
5 + n = 2
n = - 3
so, m = 5 and n = -3
Did you get it now???