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$Express the following fraction in simplest form, only using positive exponents. (-2(t^(3))^(-3))/(10t^(2)b^(6)) Answer:$

Express the following fraction in simplest form, only using positive exponents. $\newline$ $\frac{-2\left(t^{3}\right)^{-3}}{10 t^{2} b^{6}}$ $\newline$ 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{-2\left(t^{3}\right)^{-3}}{10 t^{2} b^{6}}

\newline

Answer:

Write Expression, Identify Exponents: Write down the given expression and identify negative exponents. $\newline$ The given expression is $(-2(t^{3})^{-3})/(10t^{2}b^{6})$ . We need to simplify this expression and express it with only positive exponents.
Apply Negative Exponent Rule: Apply the negative exponent rule, which states that $a^{-n} = \frac{1}{a^n}$ , to the term $(t^{3})^{-3}$ . $\newline$ $(t^{3})^{-3} = \frac{1}{(t^{3})^3}$
Simplify Exponent: Simplify the exponent $(t^{3})^{3}$ . $(t^{3})^{3} = t^{3\times3} = t^{9}$
Replace Term with Simplified Form: Replace the term $(t^{3})^{-3}$ in the original expression with its simplified form. $\frac{-2 \times \frac{1}{t^{9}}}{10t^{2}b^{6}}$
Simplify Fraction: Simplify the fraction by multiplying the numerator and denominator by $t^9$ to eliminate the negative exponent. $\newline$ $\frac{-2 \times 1}{10t^{2}b^{6} \times t^9}$
Combine Like Terms: Combine the $t$ terms in the denominator. $\frac{-2}{10t^{2+9}b^{6}}$ $\frac{-2}{10t^{11}b^{6}}$
Reduce Common Factors: Simplify the fraction by reducing common factors. $\newline$ Both the numerator and the denominator have a factor of $2$ that can be canceled out. $\newline$ $(-1)/(5t^{11}b^{6})$

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