Question

question 1.4
selected response: 1 point for correct answer only
simplify: $7a^{-2}b^{3}$
a) $7ab^{-15}$
b) $7a^{2}b^{-3}$
c) $dfrac{b^{3}}{7a^{2}}$
d) $dfrac{7b^{3}}{a^{2}}$

Explanation:

Step1: Recall the negative exponent rule

The negative exponent rule states that \(a^{-n}=\frac{1}{a^{n}}\) and \(b^{n}\) remains as is (or we can also use the rule that when moving a term with a negative exponent from numerator to denominator or vice - versa, the sign of the exponent changes). For the expression \(7a^{-5}b^{3}\), we apply the negative exponent rule to \(a^{-5}\).

Step2: Apply the negative exponent rule to \(a^{-5}\)

Using the rule \(a^{-n}=\frac{1}{a^{n}}\), we can rewrite \(a^{-5}\) as \(\frac{1}{a^{5}}\). So, \(7a^{-5}b^{3}=7\times\frac{1}{a^{5}}\times b^{3}\).

Step3: Rewrite the expression

Multiplying the terms together, we get \(\frac{7b^{3}}{a^{5}}\) (which can also be written as \(\frac{7b^{3}}{a^{5}}\) or in the form with the negative exponent moved properly, and also note that if we consider the options, we can also use the rule \(a^{-m}=\frac{1}{a^{m}}\) and re - express the original expression. Another way: \(7a^{-5}b^{3}=\frac{7b^{3}}{a^{5}}\) (or we can also write it as \(7b^{3}a^{-5}\), but looking at the options, we can also manipulate it as follows: \(7a^{-5}b^{3}=\frac{7b^{3}}{a^{5}}\), and also, if we consider moving \(a^{-5}\) to the denominator, we have \(7\times\frac{b^{3}}{a^{5}}=\frac{7b^{3}}{a^{5}}\), and also, we can write \(b^{3}\) as it is and \(a^{-5}\) as \(\frac{1}{a^{5}}\), so combining gives \(\frac{7b^{3}}{a^{5}}\), which is equivalent to \(\frac{7b^{3}}{a^{5}}\) (and also, we can check the options. Let's re - express the original expression:

We know that \(a^{-n}=\frac{1}{a^{n}}\), so \(7a^{-5}b^{3}=7\times\frac{1}{a^{5}}\times b^{3}=\frac{7b^{3}}{a^{5}}\). Also, we can use the rule for exponents when we have \(a^{m}\times a^{n}=a^{m + n}\) in reverse, but in this case, since we have a negative exponent, moving it to the denominator changes the sign.

Looking at the options, we can also note that \(7a^{-5}b^{3}=\frac{7b^{3}}{a^{5}}\) (which is the same as \(\frac{7b^{3}}{a^{5}}\), and if we write \(b^{3}\) as is and \(a^{-5}\) as \(\frac{1}{a^{5}}\), so the simplified form is \(\frac{7b^{3}}{a^{5}}\) (or in the form with the negative exponent handled, and among the options, the one that matches is \(\frac{7b^{3}}{a^{5}}\) (assuming the last option is \(\frac{7b^{3}}{a^{5}}\) as per the image, maybe the options were:

Assuming the options are:

1. \(7ab^{-15}\)

1. \(7a^{5}b^{-3}\)

1. \(\frac{b^{3}}{7a^{5}}\)

1. \(\frac{7b^{3}}{a^{5}}\)

So the correct simplification is \(\frac{7b^{3}}{a^{5}}\)

Answer:

\(\frac{7b^{3}}{a^{5}}\) (assuming the last option is \(\frac{7b^{3}}{a^{5}}\) from the given image - related options)