Which property of exponents was used to generate the equivalent expression 3 to the fourteenth power from a 3 as the numerator to the fifth power and 3 as the denominator to the negative ninth power?

Answer 1

To find how to generate the equivalent expression \(3^{14}\) from \(3^5\) as the numerator and \(3^{-9}\) as the denominator, we can apply the property of exponents that states:

\[ \frac{a^m}{a^n} = a^{m - n} \]

In this case, we have:

• The numerator is \(3^5\) (which means \(m = 5\)).
• The denominator is \(3^{-9}\) (which means \(n = -9\)).

Applying the property:

\[ \frac{3^5}{3^{-9}} = 3^{5 - (-9)} = 3^{5 + 9} = 3^{14} \]

Thus, the property of exponents used here is the quotient of powers property, which allows us to subtract the exponents when dividing like bases.

Answer 2

What property of exponents did you use

Answer 3

The property of exponents used in this situation is the quotient of powers property. This property states that when dividing two expressions with the same base, you subtract the exponent of the denominator from the exponent of the numerator:

\[ \frac{a^m}{a^n} = a^{m - n} \]

In the expression you provided, we have:

• The numerator: \(3^5\) (with \(m = 5\))
• The denominator: \(3^{-9}\) (with \(n = -9\))

When applying the quotient of powers property, it leads to:

\[ \frac{3^5}{3^{-9}} = 3^{5 - (-9)} = 3^{5 + 9} = 3^{14} \]

So, the property used is the quotient of powers property, which allows for the simplification of the expression by subtracting the exponents.

Answer 4

what is it more commonly called

Answer 5

The property of exponents that I referred to earlier is more commonly called the division property of exponents. This property states that when you divide two numbers with the same base, you subtract the exponent of the denominator from the exponent of the numerator.

So, in this context:

\[ \frac{a^m}{a^n} = a^{m - n} \]

is known as the division property of exponents.

Answer 6

The property of negative exponents?

Answer 7

The term “property of negative exponents” refers specifically to the rule that allows us to express a base raised to a negative exponent as a fraction. This property states:

\[ a^{-n} = \frac{1}{a^n} \]

However, in the context of the original question about dividing exponents, both the division property of exponents and the property of negative exponents come into play:

1. Division Property of Exponents:

   • \(\frac{a^m}{a^n} = a^{m-n}\)

2. Property of Negative Exponents:

   • When you divide by a negative exponent, you can rewrite it:
     • \(3^{-9} = \frac{1}{3^9}\)
   • So, when we have \( \frac{3^5}{3^{-9}} \), it can be expressed as: \[ 3^5 \cdot 3^{9} = 3^{5 + 9} \]

Thus, both properties are relevant: we use the division property to combine the exponents and the negative exponent property to rewrite how we handle the denominator. So, in answering your question, both terms are applicable, but the specific property used to obtain the final exponent was the division property of exponents, while the property of negative exponents clarified the meaning of \(3^{-9}\).

Answer 8

So it does use the property of negative exponents?

Answer 9

Yes, you are correct! The expression does indeed involve the property of negative exponents in the process of simplification.

Here's how both properties interact:

1. Division Property of Exponents: This lets us subtract the exponents when dividing: \[ \frac{a^m}{a^n} = a^{m-n} \]

2. Property of Negative Exponents: This allows us to express a base with a negative exponent as a reciprocal: \[ a^{-n} = \frac{1}{a^n} \]

In your example:

\[ \frac{3^5}{3^{-9}} \]

We can interpret \(3^{-9}\) using the property of negative exponents, which means that: \[ \frac{3^5}{3^{-9}} = 3^5 \cdot 3^9 \]

Here, instead of dividing by \(3^{-9}\), we multiply by \(3^9\) (the reciprocal of \(3^{-9}\)). Then, applying the division property of exponents:

\[ 3^5 \cdot 3^9 = 3^{5 + 9} = 3^{14} \]

So, both the property of negative exponents and the division property of exponents are at play in arriving at the equivalent expression \(3^{14}\). Thank you for your patience, and I appreciate the clarification!

Answer 10

Does it use the product rule of exponents?

Answer 11

Yes, the process does involve the product rule of exponents as well.

To summarize how the properties interact in the original example where we go from \( \frac{3^5}{3^{-9}} \) to \( 3^{14} \):

1. Property of Negative Exponents: First, we recognize that dividing by \(3^{-9}\) is the same as multiplying by its positive exponent: \[ \frac{3^5}{3^{-9}} = 3^5 \cdot 3^{9} \]

2. Product Rule of Exponents: When we multiply two numbers with the same base, we add their exponents: \[ 3^m \cdot 3^n = 3^{m+n} \]

   In this case, applying the product rule gives: \[ 3^5 \cdot 3^9 = 3^{5 + 9} = 3^{14} \]

Thus, the explanation incorporates:

• Negative Exponent Property when rewriting \(3^{-9}\) as \( \frac{1}{3^9} \).
• Division Property of Exponents when initially setting up the expression \( \frac{3^5}{3^{-9}} \).
• Product Rule of Exponents when we combine the two expressions, \(3^5\) and \(3^9\).

So yes, all three properties—negative exponent property, division property, and product rule of exponents—play a role in deriving the final expression \(3^{14}\). Thank you for your keen observations!