To evaluate [tex]\( 625^{-3/4} \)[/tex], let's break down the expression step by step.
1. Understand the base and exponent:
- The base of our expression is 625.
- The exponent is [tex]\(-3/4\)[/tex], which can be broken into two parts:
- The negative sign indicates a reciprocal.
- The fraction [tex]\(\frac{3}{4}\)[/tex] can be interpreted as both taking a root and a power.
2. Consider the negative exponent:
[tex]\[
x^{-a} = \frac{1}{x^a}
\][/tex]
Thus, [tex]\( 625^{-3/4} \)[/tex] can be written as:
[tex]\[
625^{-3/4} = \frac{1}{625^{3/4}}
\][/tex]
3. Interpret the fractional exponent [tex]\( \frac{3}{4} \)[/tex]:
[tex]\( 625^{3/4} \)[/tex] can be written as:
[tex]\[
625^{3/4} = (625^{1/4})^3
\][/tex]
This suggests that we first find the fourth root of 625 and then cube the result.
4. Calculate the fourth root of 625:
- We need to find a number which, when raised to the fourth power, equals 625.
- Recognize that [tex]\( 625 = 5^4 \)[/tex].
- Therefore, the fourth root of 625 is 5:
[tex]\[
625^{1/4} = 5
\][/tex]
5. Cube the fourth root:
Now we raise the fourth root to the power of 3:
[tex]\[
(625^{1/4})^3 = 5^3 = 125
\][/tex]
So, [tex]\( 625^{3/4} = 125 \)[/tex].
6. Take the reciprocal:
Finally, we take the reciprocal since the original exponent was negative:
[tex]\[
\frac{1}{625^{3/4}} = \frac{1}{125}
\][/tex]
7. Convert [tex]\( \frac{1}{125} \)[/tex] to decimal:
We know that:
[tex]\[
\frac{1}{125} = 0.008
\][/tex]
Thus, [tex]\( 625^{-3/4} = 0.008 \)[/tex].
