Sure, let's simplify the expression [tex]\( 16^{-\frac{5}{4}} \)[/tex] step-by-step.
1. Understand the base and the exponent:
The expression is in the form [tex]\( a^{b} \)[/tex] where [tex]\( a = 16 \)[/tex] and [tex]\( b = -\frac{5}{4} \)[/tex].
2. Rewrite the base as a power of a simpler number:
Notice that 16 can be expressed as a power of 2:
[tex]\[
16 = 2^4
\][/tex]
Thus, we can rewrite the expression as:
[tex]\[
16^{-\frac{5}{4}} = \left(2^4\right)^{-\frac{5}{4}}
\][/tex]
3. Apply the power of a power rule:
When you have [tex]\( \left(a^m\right)^n \)[/tex], it simplifies to [tex]\( a^{m \cdot n} \)[/tex]. Therefore:
[tex]\[
\left(2^4\right)^{-\frac{5}{4}} = 2^{4 \cdot -\frac{5}{4}}
\][/tex]
4. Multiply the exponents:
[tex]\[
4 \cdot -\frac{5}{4} = -5
\][/tex]
So the expression simplifies to:
[tex]\[
2^{-5}
\][/tex]
5. Simplify the negative exponent:
A negative exponent means that you take the reciprocal of the base raised to the positive exponent. So:
[tex]\[
2^{-5} = \frac{1}{2^5}
\][/tex]
6. Calculate the power of 2:
[tex]\[
2^5 = 32
\][/tex]
Therefore:
[tex]\[
\frac{1}{2^5} = \frac{1}{32}
\][/tex]
7. Conclusion:
The simplified value of [tex]\( 16^{-\frac{5}{4}} \)[/tex] is:
[tex]\[
16^{-\frac{5}{4}} = \frac{1}{32} = 0.03125
\][/tex]
Thus, [tex]\( 16^{-\frac{5}{4}} \)[/tex] simplifies to [tex]\( 0.03125 \)[/tex].
