Asked by Sally
Use the quotient rule to simplify the expression. Write the expression with positive exponents. Assume the variables represent nonzero real numbers.
StartFraction left parenthesis a plus 3 b right parenthesis squared Over left parenthesis a plus 3 b right parenthesis Superscript negative 6 EndFraction
StartFraction left parenthesis a plus 3 b right parenthesis squared Over left parenthesis a plus 3 b right parenthesis Superscript negative 6 EndFraction
Answers
Answered by
GPT-4o mini
To simplify the expression
\[
\frac{(a + 3b)^2}{(a + 3b)^{-6}},
\]
we can use the properties of exponents. According to the quotient rule of exponents, when you divide two expressions with the same base, you subtract the exponents:
\[
\frac{x^m}{x^n} = x^{m - n}.
\]
In our case, we have \( (a + 3b)^2 \) in the numerator and \( (a + 3b)^{-6} \) in the denominator. We can rewrite the expression applying the quotient rule:
\[
\frac{(a + 3b)^2}{(a + 3b)^{-6}} = (a + 3b)^{2 - (-6)} = (a + 3b)^{2 + 6} = (a + 3b)^{8}.
\]
Since we want to express the final answer with positive exponents, we arrive at:
\[
(a + 3b)^8.
\]
Therefore, the simplified expression is
\[
\boxed{(a + 3b)^8}.
\]
\[
\frac{(a + 3b)^2}{(a + 3b)^{-6}},
\]
we can use the properties of exponents. According to the quotient rule of exponents, when you divide two expressions with the same base, you subtract the exponents:
\[
\frac{x^m}{x^n} = x^{m - n}.
\]
In our case, we have \( (a + 3b)^2 \) in the numerator and \( (a + 3b)^{-6} \) in the denominator. We can rewrite the expression applying the quotient rule:
\[
\frac{(a + 3b)^2}{(a + 3b)^{-6}} = (a + 3b)^{2 - (-6)} = (a + 3b)^{2 + 6} = (a + 3b)^{8}.
\]
Since we want to express the final answer with positive exponents, we arrive at:
\[
(a + 3b)^8.
\]
Therefore, the simplified expression is
\[
\boxed{(a + 3b)^8}.
\]
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