Simplify the following expression to simplest form using only positive exponents. $\newline$ $\left(8 x^{18} y^{6}\right)^{-\frac{2}{3}}$ $\newline$ Answer:

Full solution

Q. Simplify the following expression to simplest form using only positive exponents.

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\left(8 x^{18} y^{6}\right)^{-\frac{2}{3}}

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Answer:

Apply negative exponent rule: Apply the negative exponent rule which states that $a^{-n} = \frac{1}{a^n}$ .
$(8x^{18}y^{6})^{-\frac{2}{3}} = \frac{1}{(8x^{18}y^{6})^{\frac{2}{3}}}$
Apply power of a power rule: Apply the power of a power rule which states that $(a^m)^n = a^{m*n}$ . $\frac{1}{(8x^{18}y^{6})^{\frac{2}{3}}} = \frac{1}{8^{\frac{2}{3}} \times x^{18\times\frac{2}{3}} \times y^{6\times\frac{2}{3}}}$
Calculate exponents: Calculate the exponents for each term. $\newline$ $\frac{1}{8^{\frac{2}{3}} \times x^{18\left(\frac{2}{3}\right)} \times y^{6\left(\frac{2}{3}\right)}} = \frac{1}{8^{\frac{2}{3}} \times x^{12} \times y^{4}}$
Simplify cube root of $8$ : Simplify the cube root of $8$ which is $2$ , and then raise it to the power of $2$ . $\newline$ $\frac{1}{(8^{\frac{2}{3}} * x^{12} * y^{4})} = \frac{1}{(2^2 * x^{12} * y^{4})}$
Calculate $2^2$ : Calculate $2^2$ . $\newline$ $\frac{1}{2^2 \times x^{12} \times y^{4}} = \frac{1}{4 \times x^{12} \times y^{4}}$
Write final expression: Write the final expression with positive exponents. $\newline$ $\frac{1}{4 \cdot x^{12} \cdot y^{4}} = \frac{1}{4x^{12}y^{4}}$