1. ## Indices Algebra

I have two questions that needs to be solved with algebra, please help, i figured out the answers to be -4 for the first and -1 for the second but i dont know the steps solving it with algebra.

2x = 1/16 Find X
(0.5)x = 2 Find X

2. ## Re: Indices Algebra

Originally Posted by TheElect
I have two questions that needs to be solved with algebra, please help, i figured out the answers to be -4 for the first and -1 for the second but i dont know the steps solving it with algebra.

2x = 1/16 Find X
(0.5)x = 2 Find X
\displaystyle \begin{align*} 2^x &= \frac{1}{16} \\ 2^x &= \frac{1}{2^4} \\ 2^x &= 2^{-4} \\ x&= 4 \\ \\ \left(0.5\right)^x &= 2 \\ \left(\frac{1}{2}\right)^x &= 2 \\ \frac{1}{2^x} &= 2 \\ 2^{-x} &= 2^1 \\ -x &= 1 \\ x &= -1 \end{align*}

3. ## Re: Indices Algebra

Thankyou, but can you please explain further into how that works? Sorry, my understanding of indices aren't that good

4. ## Re: Indices Algebra

Originally Posted by TheElect
Thankyou, but can you please explain further into how that works? Sorry, my understanding of indices aren't that good
That was every step. I don't know what else I can show you...

5. ## Re: Indices Algebra

Your problem seems to be not understanding what "indices" (I would say "exponents" which is a little more specific than "indices") mean. $x^2$ ('x squared') means 'x times x', $x^3$ means 'x times x times x', etc. Further, a negative exponent means "reciprocal" or "1 over" the value. In particular $2^4= 2(2)(2)(2)= 16$ so that $2^{-4}= \frac{1}{16}$.

Of course, $0.5= \frac{1}{2}$ so that $0.5= \left(\frac{1}{2^1}\right)= 2^{-1}$ and $(0.5)^x= (2^{-1})^x= 2^{-x}$

So the first equation says $2^x= 2^{-4}$ and comparing the two sides x= -4.

The second equation says $2^{-x}= 2^1$ and comparing the two sides gives -x= 1.

(There is a technical point here- that the exponential function is "one-to-one" which means that if f(x)= f(y) then x= y. Not all functions have that property but the exponential does. All you need to know is that you can do that for exponential functions.)

6. ## Re: Indices Algebra

4^x = 4^y
Can you "see" that x=y?
Let x=2: 4^2 = 16; so 4^y = 16; so y HAS TO equal x : capish?

7. ## Re: Indices Algebra

I think I understand now, thanks all of you

8. ## Re: Indices Algebra

Definition of exponentiation: b^m = b * b * b * … * b (b times itself m times)
For example: b^4 × b^3 =
( b × b × b × b) × ( b × b × b) = (definition of exponentiation)
b × b × b × b × b × b × b = (associative property of multiplication)
b^7 (definition of exponentiation)
If b^m=b^n
b^m/b^n=1
b^(m-n)=1
m-n=0
Hence, m=n