Question

exponents

1. rewrite this expression so that it contains only positive exponents. be sure to simplify.

\\(\frac{a^{-3}b^{-8}}{b^{3}c^{-4}}\\)

Explanation:

Step1: Recall the negative exponent rule

The negative exponent rule states that \(x^{-n}=\frac{1}{x^{n}}\) and \(\frac{1}{x^{-n}} = x^{n}\). We will apply this rule to each term with a negative exponent.
For the numerator: \(a^{-3}=\frac{1}{a^{3}}\) and \(b^{-8}=\frac{1}{b^{8}}\), so the numerator \(a^{-3}b^{-8}=\frac{1}{a^{3}b^{8}}\).
For the denominator: \(c^{-4}=\frac{1}{c^{4}}\), so \(\frac{1}{c^{-4}}=c^{4}\), and the denominator \(b^{3}c^{-4}=\frac{b^{3}}{c^{4}}\).
Now our expression is \(\frac{\frac{1}{a^{3}b^{8}}}{\frac{b^{3}}{c^{4}}}\).

Step2: Divide by a fraction

Dividing by a fraction is the same as multiplying by its reciprocal, so \(\frac{\frac{1}{a^{3}b^{8}}}{\frac{b^{3}}{c^{4}}}=\frac{1}{a^{3}b^{8}}\times\frac{c^{4}}{b^{3}}\).

Step3: Multiply the fractions and use exponent rule for multiplication

When multiplying fractions, we multiply the numerators and the denominators. Also, when multiplying like bases with exponents, we use the rule \(x^{m}\times x^{n}=x^{m + n}\).
For the \(b\) terms: \(b^{8}\times b^{3}=b^{8 + 3}=b^{11}\).
So \(\frac{1\times c^{4}}{a^{3}\times b^{8}\times b^{3}}=\frac{c^{4}}{a^{3}b^{11}}\).

Answer:

\(\frac{c^{4}}{a^{3}b^{11}}\)