Question

simplify using the laws of exponents: \\(dfrac{15a^{10}b^5}{5a^8b^2}\\)

Explanation:

Step1: Simplify the coefficient

First, simplify the numerical coefficient. We have \(\frac{15}{5}\). Since \(15\div5 = 3\), the coefficient simplifies to \(3\).

Step2: Simplify the \(a\)-terms using the quotient rule for exponents

The quotient rule for exponents is \(\frac{a^m}{a^n}=a^{m - n}\). For the \(a\)-terms, we have \(\frac{a^{10}}{a^{8}}\). Applying the quotient rule, \(m = 10\) and \(n = 8\), so \(a^{10-8}=a^{2}\).

Step3: Simplify the \(b\)-terms using the quotient rule for exponents

For the \(b\)-terms, we have \(\frac{b^{5}}{b^{2}}\). Using the quotient rule for exponents with \(m = 5\) and \(n = 2\), we get \(b^{5 - 2}=b^{3}\).

Step4: Combine the simplified parts

Now, combine the simplified coefficient, \(a\)-term, and \(b\)-term. We multiply them together: \(3\times a^{2}\times b^{3}=3a^{2}b^{3}\).

Answer:

\(3a^{2}b^{3}\)