Easy Exponents.

Algebra Level 2

If the value of the following expression $\dfrac{2^{-5}\times 5^6\times 64\times 6561}{30^8\times 2^{-2}\times 5^2}$

can be expressed as $5\times 10^a$ , then find the value of $a$ .

The answer is -5.

4 solutions

Anna Anant
Dec 15, 2014

We simplify this expression to have just power of 2, 3 and 5. 64=2^6 6561= 9^4=3^8

30^8=(2x5x10)^8=2^8 x 3^8 x 5^8

the expression will be as : E=[2^(-5) x 5^6 x 2^6 x 3^8] / [2^8 x 3^8 x 5^8 x 2^(-2) x 5^2] =[2^(-5+6-8+2) x 3^(8-8) x 5^(6-8-2)] =2^(-5) x 3^0 x 5^(-4)

=2^(-5) x 5^(-4)

5 x 10^a = 2^(-5) x 5^(-4) 5 x 5^a x 2^a = 2^(-5) x 5^(-4) 5^(a+1) x 2^a = 2^(-5) x 5^(-4) 5^(a+1+4)x2^(a+5)=1 a+5=0 a=-5

Mhar Ariz Marino
Dec 19, 2014

(simplify simplify simplify simplify simplify)

Nain Tara
Dec 15, 2014

just simplify it!! and that's how u will get it !!

Pralhad Das
Dec 14, 2014

just simplify it!! and that's how u will get it !!

I think the point is to do it without a calculator... hello?

Joshua Wakefield - 6 years, 6 months ago

It can be done algebraically. Here' how:

64 = 2^6

6561 = 81^2 = 9^4 = 3^8

30^8 = 2^8 * 3^8 * 5^8

5*10^a = (2^-5 * 5^6 * 2^6 * 3^8) / (2^8 * 3^8 * 5^8 * 2^-2 *5^2)

Combine the powers of 2 and 5 into the numerator, cancel the two 3^8's, and expand the left hand side of the equation:

2^a * 5^a * 5 = 2^(-5+6-8+2) * 5^(6-8-2)

2^a * 5^(a+1) = 2^-5 * 5^-4

Divide left hand side by entire right hand side and simplify:

2^(a+5) * 5^(a +1+4) = 1

2^(a+5) * 5^(a+5) = 10^(a+5) = 1 = 10^0

10^(a+5) = 10^0

Exponents must be equal, so solve for a:

a + 5 = 0

a = -5