Sure! Let's work through the problem step-by-step:
We start with the expression:
[tex]$
\left(\frac{5}{2}\right)^{-2}
$[/tex]
### Step 1: Rewrite using positive exponents
A negative exponent indicates that we take the reciprocal of the base and then apply the positive version of that exponent. Therefore, we can rewrite the expression as:
[tex]$
\left(\frac{5}{2}\right)^{-2} = \frac{1}{\left(\frac{5}{2}\right)^2}
$[/tex]
### Step 2: Calculate the positive exponent
Now we need to calculate [tex]\(\left(\frac{5}{2}\right)^2\)[/tex]. Squaring a fraction means squaring both the numerator and the denominator separately:
[tex]$
\left(\frac{5}{2}\right)^2 = \frac{5^2}{2^2} = \frac{25}{4}
$[/tex]
### Step 3: Rewrite the expression using the calculated value
We now substitute the squared value back into the expression:
[tex]$
\frac{1}{\left(\frac{5}{2}\right)^2} = \frac{1}{\frac{25}{4}}
$[/tex]
### Step 4: Simplify the expression
To simplify [tex]\(\frac{1}{\frac{25}{4}}\)[/tex], we take the reciprocal of [tex]\(\frac{25}{4}\)[/tex]:
[tex]$
\frac{1}{\frac{25}{4}} = \frac{4}{25}
$[/tex]
So the expression [tex]\(\left(\frac{5}{2}\right)^{-2}\)[/tex] simplifies to:
[tex]$
\frac{4}{25}
$[/tex]
Therefore, the final answer is:
[tex]$
\left(\frac{5}{2}\right)^{-2} = \frac{4}{25} \approx 0.16
$[/tex]
