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$Express the following fraction in simplest form, only using positive exponents. (-4b^(-1))/((-3b^(3))^(5)) Answer:$

Express the following fraction in simplest form, only using positive exponents. $\newline$ $\frac{-4 b^{-1}}{\left(-3 b^{3}\right)^{5}}$ $\newline$ Answer:

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Multiplication with rational exponents

Full solution

Q. Express the following fraction in simplest form, only using positive exponents.

\newline

\frac{-4 b^{-1}}{\left(-3 b^{3}\right)^{5}}

\newline

Answer:

Simplify Denominator: Simplify the denominator. $\newline$ The denominator is $(-3b^{3})^5$ . When raising a power to a power, you multiply the exponents. $\newline$ $(-3b^{3})^5 = (-3)^5 \times (b^3)^5$ $\newline$ $= -243 \times b^{3\times5}$ $\newline$ $= -243 \times b^{15}$
Rewrite Negative Exponent: Rewrite the negative exponent in the numerator as a positive exponent. $\newline$ The numerator is $-4b^{-1}$ . A negative exponent means that the base is on the wrong side of the fraction line, so we flip it to the other side to make the exponent positive. $\newline$ $-4b^{-1} = -\frac{4}{b}$
Combine Numerator and Denominator: Combine the rewritten numerator and the simplified denominator. $\newline$ Now we have $-\frac{4}{b}$ divided by $-243 \times b^{15}$ . $\newline$ \left(-\frac{\(4\)}{b}\right) / \left(\(-243 \times b^{ $15$ }\right) = \left(-\frac{ $4$ }{b}\right) \times \left(\frac{ $1$ }{ $-243$ \times b^{ $15$ }}\right)
Multiply Numerators and Denominators: Multiply the numerators and denominators. $\newline$ When multiplying fractions, you multiply the numerators together and the denominators together. $\newline$ $(-4 \times 1) / (b \times -243 \times b^{15}) = -4 / (-243 \times b^{16})$
Simplify Fraction: Simplify the fraction by dividing by the common factor and making the exponent positive. $\newline$ We can divide both the numerator and the denominator by $-1$ to make the fraction positive. $\newline$ $-4 / (-243 * b^{16}) = 4 / (243 * b^{16})$

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