p) $$ 2\left(\frac{1}{2}\right)^{-x}=5^{-2} \cdot 5^{2} $$ $$ \frac{2\left(\frac{1}{2}\right)^{-x}}{2}=\frac{5^{-2} \cdot 5^{2}}{2} $$

Question

UpStudy AI · Accepted Answer

We start with the given equation:

2·(½)^(–x) = 5^(–2) · 5^2

Step 1. Simplify the right-hand side using the exponent rule a^(m)·a^(n) = a^(m+n):

5^(–2)·5^2 = 5^(–2 + 2) = 5^0 = 1

Now the equation becomes:

2·(½)^(–x) = 1

Step 2. Divide both sides by 2:

(½)^(–x) = 1/2

Step 3. Rewrite the left-hand side by using the property (a^b)^(–1) = 1/(a^b), or note that (½)^(–x) is equivalent to 2^x. More directly,

(½)^(–x) = (2)^(x)         (Changing the sign of the exponent “flips” the base)

Now we have:

2^x = 1/2

Step 4. Express 1/2 as a power of 2:

1/2 = 2^(–1)

Thus, the equation is:

2^x = 2^(–1)

Since the bases are the same, equate the exponents:

x = –1

So, the solution is x = –1.
The solution is $$ x = -1 $$.

p) \( 2\left(\frac{1}{2}\right)^{-x}=5^{-2} \cdot 5^{2} \) \( \frac{2\left(\frac{1}{2}\right)^{-x}}{2}=\frac{5^{-2} \cdot 5^{2}}{2} \)