Question

write the expression with positive exponents only.
\\(\frac{x^{-6}y^{9}}{z^{-7}}\\)

\\(\frac{x^{-6}y^{9}}{z^{-7}} = \square\\)
(type exponential notation with positive exponents. simplify your answer.)

Explanation:

Step1: Recall the negative exponent rule

The negative exponent rule states that \( a^{-n}=\frac{1}{a^{n}} \) and \( \frac{1}{a^{-n}} = a^{n} \) for any non - zero real number \( a \) and positive integer \( n \).

For the term \( x^{-6} \) in the numerator, using the rule \( a^{-n}=\frac{1}{a^{n}} \), we can rewrite \( x^{-6} \) as \( \frac{1}{x^{6}} \), but we want to move it to the denominator to make the exponent positive.

For the term \( z^{-7} \) in the denominator, using the rule \( \frac{1}{a^{-n}}=a^{n} \), we can rewrite \( \frac{1}{z^{-7}} \) as \( z^{7} \).

Step2: Rewrite the expression

The original expression is \( \frac{x^{-6}y^{9}}{z^{-7}} \).

Using the negative exponent rules:

• \( x^{-6}=\frac{1}{x^{6}} \), so when we have \( x^{-6} \) in the numerator, to make the exponent positive, we move it to the denominator: \( \frac{y^{9}}{x^{6}z^{-7}} \)
• Then, for \( z^{-7} \) in the denominator, we use \( \frac{1}{z^{-7}} = z^{7} \), so we move \( z \) from the denominator to the numerator with a positive exponent.

So \( \frac{x^{-6}y^{9}}{z^{-7}}=\frac{y^{9}z^{7}}{x^{6}} \)

Answer:

\(\frac{y^{9}z^{7}}{x^{6}}\)