Sure, let's solve the expression [tex]\((\sqrt[4]{a})^{-3}\)[/tex] step by step.
1. Understand the notation:
The expression [tex]\(\sqrt[4]{a}\)[/tex] represents the fourth root of [tex]\(a\)[/tex]. This can also be written using exponent notation as [tex]\(a^{\frac{1}{4}}\)[/tex].
2. Rewrite the expression:
So, [tex]\((\sqrt[4]{a})^{-3}\)[/tex] can be rewritten as:
[tex]\[
\left(a^{\frac{1}{4}}\right)^{-3}
\][/tex]
3. Apply the power rule:
Using the power rule [tex]\((x^m)^n = x^{m \cdot n}\)[/tex], we can simplify the expression by multiplying the exponents:
[tex]\[
a^{\frac{1}{4} \cdot (-3)}
\][/tex]
4. Multiply the exponents:
Now, multiply the exponents [tex]\(\frac{1}{4} \cdot (-3)\)[/tex]:
[tex]\[
\frac{1}{4} \cdot (-3) = -\frac{3}{4}
\][/tex]
5. Rewrite the expression:
Therefore, the expression [tex]\(\left(a^{\frac{1}{4}}\right)^{-3}\)[/tex] simplifies to:
[tex]\[
a^{-\frac{3}{4}}
\][/tex]
So, the final simplified form of the given expression [tex]\((\sqrt[4]{a})^{-3}\)[/tex] is:
[tex]\[
a^{-\frac{3}{4}}
\][/tex]
