Question

simplify. express your answer using a single exponent. ((y^{3})^{-6})

Explanation:

Step1: Recall exponent rule

When raising a power to a power, we multiply the exponents: \((a^m)^n = a^{m\times n}\).

Step2: Apply the rule to \((y^3)^{-6}\)

Here, \(a = y\), \(m = 3\), and \(n=-6\). So we multiply the exponents: \(3\times(-6)= - 18\). Thus, \((y^3)^{-6}=y^{-18}\) or we can also write it as \(\frac{1}{y^{18}}\) (using the rule \(a^{-n}=\frac{1}{a^n}\)), but since the problem says "express your answer using a single exponent", \(y^{-18}\) is also acceptable. But usually, if we consider positive exponents, \(\frac{1}{y^{18}}\) is also with a single exponent (the exponent of \(y\) in the denominator is \(18\)). However, following the rule of multiplying exponents directly: \((y^3)^{-6}=y^{3\times(-6)} = y^{-18}\) or \(\frac{1}{y^{18}}\).

Answer:

\(y^{-18}\) (or \(\frac{1}{y^{18}}\))