Question

What is the value of AAA when we rewrite 2^{x-6}+2^{x}2 x−6 +2 x 2, start superscript, x, minus, 6, end superscript, plus, 2, start superscript, x, end superscript as A\cdot 2^{x}A⋅2 x A, dot, 2, start superscript, x, end superscript ?

Answer 1

The value of AAA is 2^(x-6) + 2^(3x-6) + 2^(2x).

Step-by-step explanation:

The student is asking for the value of {AAA} when the expression {2^(x-6) + 2^x2 x-6 + 2 x 2x} is rewritten as {A⋅2^x⋅A⋅2x}. To find the value, we need to simplify the expression.

Using the product rule of exponents, we can rewrite the expression as {2^(x-6) + 2^(x)2^(x-6) + 2^(2x)}. Now we can combine the like terms by adding the exponents. This gives us {2^(x-6) + 2^(2x)2^(x-6) + 2^(2x)}.

By applying the power rule, {a^m+a^n = a^(m+n)}, we can simplify further to get {2^(x-6) + 2^(2x+x-6) + 2^(2x)}. This can be written as {2^(x-6) + 2^(3x-6) + 2^(2x)}. Therefore, the value of {AAA} is {2^(x-6) + 2^(3x-6) + 2^(2x)}.

Answer 2

`A=(65)/(64)\ or\ A=1.015625`

Step-by-step explanation:

Given:

The expression is given as:

`2^(x-6)+2^(x)`

The equivalent expression to the above expression is given as:

`A\cdot 2^(x)`

Now, simplifying the original expression using the law of indices:

`a^(m-n)=(a^m)/(a^n)`

So,
`2^(x-6)=(2^x)/(2^6)`. The expression becomes:

`=(2^x)/(2^6)+2^x`

Now,
`2^x` is a common factor to both the terms, so we factor it out. This gives,

`=2^x((1)/(2^6)+1)\\\\=2^x((1)/(64)+1)\\\\=2^x((1)/(64)+(64)/(64))\\\\=2^x((1+64)/(64))\\\\=2^x((65)/(64))\\\\=((65)/(64))\cdot 2^x`

Now, on comparing the simplified form with the equivalent expression, we conclude:

`A\cdot2^x=((65)/(64))\cdot 2^x\\\\A=(65)/(64)\ or\ 1.015625`

Therefore, the value of 'A' is
`A=(65)/(64)\ or\ A=1.015625`