Certainly! Let's rewrite [tex]\(\left(\frac{5}{2}\right)^{-3}\)[/tex] without using an exponent. Here's a step-by-step breakdown of the solution:
1. Starting Expression:
[tex]\[\left(\frac{5}{2}\right)^{-3}\][/tex]
2. Apply Negative Exponent Rule:
A negative exponent indicates that we can take the reciprocal of the base and change the sign of the exponent to positive.
[tex]\[\left(\frac{5}{2}\right)^{-3} = \left(\frac{2}{5}\right)^{3}\][/tex]
3. Evaluating the Positive Exponent:
Now, let's compute [tex]\(\left(\frac{2}{5}\right)^{3}\)[/tex].
[tex]\[\left(\frac{2}{5}\right)^{3} = \frac{2^3}{5^3}\][/tex]
4. Calculate Powers:
Evaluate the powers in the numerator and the denominator.
[tex]\[2^3 = 8\][/tex]
[tex]\[5^3 = 125\][/tex]
5. Form the Fraction:
Substitute the computed values back into the fraction.
[tex]\[\frac{2^3}{5^3} = \frac{8}{125}\][/tex]
Therefore, without using an exponent, [tex]\(\left(\frac{5}{2}\right)^{-3} = \frac{8}{125}\)[/tex].
Finally, converting this fraction to a decimal gives:
[tex]\[\frac{8}{125} = 0.064\][/tex]
So, the value is [tex]\(0.064\)[/tex].
