Question

a45 - exponents power rule
score: 5/10 answered: 5/10
question 6
simplify: \\(\frac{a^{9}b \cdot a^{8}b^{2}}{(a^{2}b^{8})^{2}}\\)
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Explanation:

Step1: Simplify numerator (product rule)

For the numerator \(a^{9}b \cdot a^{8}b^{2}\), use the product rule of exponents \(x^m \cdot x^n = x^{m + n}\) for like bases.
For \(a\): \(a^{9} \cdot a^{8}=a^{9 + 8}=a^{17}\)
For \(b\): \(b^{1} \cdot b^{2}=b^{1 + 2}=b^{3}\)
So numerator becomes \(a^{17}b^{3}\).

Step2: Simplify denominator (power of a product rule)

For the denominator \((a^{2}b^{8})^{2}\), use the power of a product rule \((xy)^n = x^n y^n\) and power of a power rule \((x^m)^n = x^{m \cdot n}\).
For \(a\): \((a^{2})^{2}=a^{2 \cdot 2}=a^{4}\)
For \(b\): \((b^{8})^{2}=b^{8 \cdot 2}=b^{16}\)
So denominator becomes \(a^{4}b^{16}\).

Step3: Divide numerator by denominator (quotient rule)

Now we have \(\frac{a^{17}b^{3}}{a^{4}b^{16}}\). Use the quotient rule of exponents \(x^m \div x^n = x^{m - n}\) for like bases.
For \(a\): \(a^{17} \div a^{4}=a^{17 - 4}=a^{13}\)
For \(b\): \(b^{3} \div b^{16}=b^{3 - 16}=b^{-13}\)
Recall that \(b^{-n}=\frac{1}{b^n}\), so \(b^{-13}=\frac{1}{b^{13}}\). Thus, the expression simplifies to \(\frac{a^{13}}{b^{13}}\) or \(a^{13}b^{-13}\).

Answer:

\(\boldsymbol{\frac{a^{13}}{b^{13}}}\) (or \(a^{13}b^{-13}\))