Question

simplify. express your answer using exponents.
$(b^{-6}c^{-2}d^{2})^{-5}$

Explanation:

Step1: Apply power of a product rule

When raising a product to a power, we raise each factor to that power. So, \((b^{-6}c^{-2}d^{2})^{-5}=b^{-6\times(-5)}c^{-2\times(-5)}d^{2\times(-5)}\)

Step2: Simplify each exponent

Calculate each product of exponents: \(b^{30}c^{10}d^{-10}\) (or we can write \(d^{-10}\) as \(\frac{1}{d^{10}}\), but since the problem says to express using exponents, negative exponents are acceptable too. However, usually we prefer positive exponents in the numerator, so we can rewrite \(d^{-10}\) as \(\frac{1}{d^{10}}\), but actually, the rule is \(a^{m\times n}\) when \((a^m)^n\), so let's correct the step2: Wait, no, the power of a power rule is \((a^m)^n = a^{m\times n}\). So for \(b^{-6}\) raised to \(-5\): \(-6\times -5 = 30\), so \(b^{30}\); for \(c^{-2}\) raised to \(-5\): \(-2\times -5 = 10\), so \(c^{10}\); for \(d^{2}\) raised to \(-5\): \(2\times -5 = -10\), so \(d^{-10}\). But we can also express \(d^{-10}\) as \(\frac{1}{d^{10}}\), but the problem says "express your answer using exponents", so negative exponents are allowed. But typically, we write with positive exponents in the numerator, so we can rewrite \(d^{-10}\) as \(\frac{1}{d^{10}}\), but actually, the simplified form with positive exponents would be \(\frac{b^{30}c^{10}}{d^{10}}\). Wait, let's redo the steps correctly:

Step1: Apply \((xy)^n = x^n y^n\) and \((x^m)^n = x^{m\times n}\)

So \((b^{-6}c^{-2}d^{2})^{-5} = (b^{-6})^{-5} \times (c^{-2})^{-5} \times (d^{2})^{-5}\)

Step2: Simplify each term

For \((b^{-6})^{-5}\): using \((x^m)^n = x^{m\times n}\), we get \(b^{-6\times -5} = b^{30}\)

For \((c^{-2})^{-5}\): \(c^{-2\times -5} = c^{10}\)

For \((d^{2})^{-5}\): \(d^{2\times -5} = d^{-10}\)

Now, \(d^{-10} = \frac{1}{d^{10}}\), so combining all terms, we have \(b^{30}c^{10}d^{-10} = \frac{b^{30}c^{10}}{d^{10}}\) (by the rule \(a^{-n} = \frac{1}{a^n}\))

Answer:

\(\frac{b^{30}c^{10}}{d^{10}}\) (or \(b^{30}c^{10}d^{-10}\) but the standard simplified form with positive exponents in the numerator is \(\frac{b^{30}c^{10}}{d^{10}}\))