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

(100x^(12)y^(-14))^(-(1)/(2))
Answer:

Simplify the following expression to simplest form using only positive exponents. $\newline$ $\left(100 x^{12} y^{-14}\right)^{-\frac{1}{2}}$ $\newline$ Answer:

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Multiplication with rational exponents

Full solution

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

\newline

\left(100 x^{12} y^{-14}\right)^{-\frac{1}{2}}

\newline

Answer:

Apply Negative Exponent Rule: Apply the negative exponent rule to the entire expression. $\newline$ The negative exponent rule states that $a^{-n} = \frac{1}{a^n}$ . We will apply this rule to the entire expression $(100x^{12}y^{-14})^{-\frac{1}{2}}$ .
Distribute Exponent to Factors: Distribute the exponent to each factor inside the parentheses. $\newline$ When an exponent is applied to a product, it is distributed to each factor. So, we will apply the exponent $-\frac{1}{2}$ to $100$ , $x^{12}$ , and $y^{-14}$ separately. $\newline$ $(100x^{12}y^{-14})^{-\frac{1}{2}} = 100^{-\frac{1}{2}} \times x^{12\cdot(-\frac{1}{2})} \times y^{-14\cdot(-\frac{1}{2})}$
Simplify Each Part: Simplify each part of the expression. $\newline$ Now we simplify each part of the expression: $\newline$ $100^{-(1)/(2)} = \frac{1}{100^{1/2}} = \frac{1}{10}$ because the square root of $100$ is $10$ . $\newline$ $x^{12*(-(1)/(2))} = x^{-6} = \frac{1}{x^6}$ because of the negative exponent rule. $\newline$ $y^{-14*(-(1)/(2))} = y^{7}$ because the negatives cancel out and we are left with a positive exponent.
Combine Simplified Parts: Combine the simplified parts. $\newline$ Now we combine the simplified parts to get the final expression: $\newline$ $(\frac{1}{10}) \times (\frac{1}{x^6}) \times y^{7}$
Write Final Expression: Write the final expression with positive exponents only. $\newline$ To ensure all exponents are positive, we keep $y^{7}$ in the numerator and move $(x^6)$ to the denominator: $\newline$ Final expression: $(y^{7})/(10x^{6})$

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